These appeals require us to determine whether a judge’s failure to recuse himself from criminal cases in which recusal was required necessitates vacating the resulting judgments and sentences. We conclude that, in these cases, it does not.
I.
Judge Michael K. Moore of the Southern District of Florida was notified in November 1992 that he was the subject of a federal grand jury investigation in the Eastern District of New York.1 Eleven *811months later, in October 1993, the investí-gation was reported in two Miami newspapers. Immediately after these stories appeared, Judge Moore recused himself sua sponte from all cases in which the United States was a party.
The appellees in these cases are criminal defendants who had been tried and/or sentenced by Judge Moore between November 1992 and October 1993. Each moved for a new trial and/or sentencing hearing on the ground that Judge Moore should have recused himself from their cases in November 1992, when he first learned of the investigation, rather than in October 1993, when the investigation became public. The defendants’ motions were assigned to Chief Judge William C. O’Kelley of the Northern District of Georgia,2 who concluded that Judge Moore should have recused himself in November 1992 and that the appropriate remedy for his failure to do so was vacatur. He therefore granted the motions. The Government appealed, and a panel of this court affirmed. We granted rehearing en banc.3
II.
We first consider the district court’s holding that Judge Moore violated 28 U.S.C. § 455(a)4 by failing to recuse himself from presiding at the defendants’ trials and/or sentencing hearings. See United States v. Garrudo, 869 F.Supp. 1574, 1581 (S.D.Fla.1994).5 The panel opinion affirmed the district court’s holding on this issue, but that opinion was vacated when we granted rehearing en bane in this case. See United States v. Cerceda, 139 F.3d 847, 852-55 (11th Cir.), vacated, 161 F.3d 652 (11th Cir.1998). Because the judges of the en banc court are equally divided with respect to whether Judge Moore violated section 455(a), the *812district court’s holding on this particular issue is affirmed by operation of law. See Reshard v. Britt, 839 F.2d 1499 (11th Cir.1988) (en banc) (affirming district court’s order in its entirety by an equally divided court); see also Herweg v. Ray, 619 F.2d 1265 (8th Cir.1980) (en banc), rev’d on other grounds, 455 U.S. 265, 102 S.Ct. 1059, 71 L.Ed.2d 137 (1982) (affirming district court’s judgment in part by an equally divided court and reversing it in part); Pennsylvania v. O’Neill, 473 F.2d 1029 (3d Cir.1973) (en banc) (affirming district court’s order in part by an equally divided court and vacating it in part). The panel opinion remains vacated.6
III.
In Liljeberg v. Health Services Acquisition Corp., 486 U.S. 847, 864, 108 5.Ct. 2194, 2205, 100 L.Ed.2d 855 (1988), the Supreme Court applied a three-factor test to determine whether a judicial action taken in violation of section 455(a) should be remedied by vacatur pursuant to Fed. R.Civ.P. 60(b).7 This test requires a court to consider: “[1] the risk of injustice to the parties in the particular case, [2] the risk that the denial of relief will produce injustice in other cases, and [3] the risk of undermining the public’s confidence in the judicial process.” Id.; see also Parker v. Connors Steel Co., 855 F.2d 1510, 1526 (11th Cir.1988). In determining that it was appropriate to vacate the judgments and/or sentences in the defendants’ cases pursuant to Fed.R.Crim.P. 33,8 the district court cited the Liljeberg factors but made no factual findings under any of the factors. Instead, the court simply concluded that “the third factor identified by the Supreme Court in Liljeberg is sufficient by itself to warrant a new trial.” Garrudo, 869 F.Supp. at 1582. It is appropriate, therefore, for us to consider de norvo the question of whether vacatur is warranted in these cases. See City of Tuscaloosa v. Harcros Chems., Inc., 158 F.3d 548, 556 (11th Cir.1998) (“A district court by definition abuses its discretion when it makes an error of law.”) (quoting Koon v. United States, 518 U.S. 81, 100, 116 S.Ct. 2035, 2047, 135 L.Ed.2d 392 (1996)). In so doing, we apply the three Liljeberg factors seriatim.
A.
Our consideration of the first Lil-jeberg factor — risk of injustice to the parties in the particular case — entails a two-element inquiry.9 A court applying this *813factor must consider not only the risk of injustice to the parties from any potential partiality or bias on the part of the judge, but also the risk of injustice posed by the remedy of vacatur itself. Each of these elements warrants separate exposition.
1.
Under the first Liljeberg factor, the party seeking vacatur bears the burden of proving that potential bias on the part of the judge presented a risk of injustice to it. A mere showing that the impartiality of the judge might reasonably be questioned clearly is not sufficient to carry this burden; such a showing follows by definition from a court’s finding that the judge violated section 455(a). See Liljeberg, 486 U.S. at 862, 108 S.Ct. at 2203-04. On the other hand, the party seeking vaca-tur is not required to prove that the judge’s potential bias actually prejudiced it by showing, for example, that certain rulings of the judge were erroneous and that the errors were in some way attributable to the judge’s potential bias. Among other difficulties, such a requirement often would place a court of appeals in the problematic position of determining whether the rulings indicated by the party were in fact erroneous even though the merits of the party’s ease were not properly before it. These uninformed or ill-considered determinations of error would have drastic consequences for subsequent appellate review of the party’s case on the merits, and might have the effect of foreclosing such review altogether.
Instead, the following two considerations should guide the court in determining whether the party seeking vacatur has met its burden of proving that the potential bias on the part of the judge represented a risk of injustice to it. First, the reviewing court should consider whether the party seeking vacatur has pointed to particular circumstances that may indicate a risk of injustice to that party.10 Second, the court should consider the seriousness of the violation of section 455(a) that is involved.11
*814Applying these two considerations to the cases before us, we conclude that the defendants have not carried their burden under the first Liljeberg factor. In their presentations to the district court- in support of their motions for new trial, and in their arguments to this court, the defendants have not pointed out any particular circumstances indicating a risk of injustice to them.12 With respect to the seriousness of the section 455(a) violation, we do of course recognize that the district court concluded that there was such a violation, and that this court has affirmed that conclusion by an equally divided court. However, after careful consideration of the totality of the circumstances, we now hold that the violation thus established was neither egregious nor clear to the judge. In arriving at this conclusion, we have considered the totality of the circumstances, including, inter alia, the nature of the violation, the egregiousness thereof, the clarity of the violation, and the reasonableness of the judge’s lack of awareness that he was in violation of section 455(a).13
We leave open the possibility that in a rare case involving an extremely serious violation of section 455(a), a court might find that the party seeking vacatur has carried its burden under the first Liljeberg factor, even if the party has pointed to no particular circumstances indicating a risk of injustice. But this is not such a case. Indeed, six of the twelve judges on this en banc court believe there was no violation of section 455(a) at all, but, assuming a violation, would join the other judges comprising the majority on the remedy issue in concluding that the violation was neither egregious nor readily apparent.
2.
The party opposing vacatur also has a burden to carry under the first Liljeberg factor. This party bears the burden of proving that the remedy of vaca-tur itself poses a risk of injustice to it. See Liljeberg, 486 U.S. at 868-69, 108 S.Ct. at 2207 (considering whether it would be “unfair to deprive the prevailing party of its judgment,” and finding that “neither [of the prevailing parties] has made a showing of special hardship by reason of their reliance on the original judgment”). In order to determine the risk of injustice that va-catur poses to the Government in these cases, we consider separately the defendants who were granted new trials and those who were granted new sentencing hearings.
We agree with the Government’s argument that the remedy of providing new trials to the defendants who have requested them poses a significant risk of injustice to it. The Government certainly would spend significant amounts of time and money in retrying each of these defendants. Resources devoted to retrials in all of these cases would have to be diverted from other cases with the ultimate result that some crime will go unpunished. See, e.g., J.E.B. v. Alabama, 511 U.S. 127, 159, 114 S.Ct. 1419, 1437, 128 L.Ed.2d 89 (1994) (Scalia, J., dissenting) (“a retrial will do nothing but divert the [government’s] resources, allowing either petitioner or some other malefactor to go free.”). Moreover, the long delay between the defendants’ original and new trials could seriously compromise the Government’s ability to re-prosecute the defendants effectively. This problem is particularly acute in certain cases. With regard to the case of *815United States v. De La Mata, No. 95-4630, for example, the government would face great hardship if forced to conduct a new trial both because of the complexity of the case (a 78 count, complex white-collar prosecution the trial of which lasted two- and-a-half months) and because a key Government witness was 84 years old and in bad health at the time of the first trial. It is questionable whether that witness — now well over 90 years old — would be capable of again providing the testimony crucial to the Government’s case.
Turning to the defendants who received new sentencing hearings, we note that the monetary and temporal costs to the Government from holding such hearings are relatively low. In addition, the Government has not shown that the lengthy delay between the defendants’ original and new sentencing hearings would present a special hardship to it in these particular cases. We conclude, therefore, that the risk of injustice to the Government from granting these resentencing hearings is slight. See Foster v. United States, 615 A.2d 213, 220 (D.C.1992); Belton v. United States, 581 A.2d 1205, 1215 (D.C.1990).
B.
Next, we examine Liljeberg’s second factor — the risk that the denial of relief will produce injustice in other cases. This factor weighs in favor of vacating the judgment when so doing would “encourag[e] a judge or litigant to more carefully examine possible grounds for disqualification and to promptly disclose them when discovered.” Liljeberg, 486 U.S. at 868, 108 S.Ct. at 2206. In this case, vacating the defendants’ convictions and sentences is not nee-essary to deter judges from violating section 455(a), because the Eleventh Circuit Judicial Council has already minimized the risk that similar violations will occur in the future. In September 1996, the Judicial Council adopted a protocol that established guidelines for recusal when a judge is notified that he is the subject or target of a criminal investigation.14 This protocol should provide substantial guidance to judges who find themselves in a situation similar to that of Judge Moore.
Although the protocol does not completely eliminate the risk that other judges will act as Judge Moore did, the risk that they will do so is low enough that vacating the defendants’ convictions and sentences would have only a minimal deterrent effect. Instead, the district court’s ruling that Judge Moore violated section 455(a), which we (albeit by an equally divided court) affirm, should sufficiently impress upon judges the need to identify and disclose potential grounds for disqualification. Thus, we conclude that there is little risk that denying relief here will produce injustice in other cases.
C.
Finally, we examine the third Liljeberg factor — the risk of undermining the public’s confidence in the judicial process. Although every violation of section 455(a) creates a risk that the public will lose faith in the judicial system, vacating the trial court’s judgment to remedy the violation sometimes increases rather than decreases that risk. This is such a case. As noted above, we hold that the section 455(a) violation, found by the district court and affirmed by this evenly divided en banc *816court, is neither egregious nor clear cut. The closeness of the violation issue, and its lack of clarity to Judge Moore, factors into the public confidence calculus. Moreover, the defendants have failed to establish any significant possibility that they suffered any harm because of the circumstances that underlie the section 455(a) issue. For these reasons, the public would lose confidence in the judicial process if the judgments were vacated, because the parties and the courts would be forced to relitigate the case even though the proceedings leading to those judgments seemed completely fair.15
This conclusion is particularly valid with respect to the defendants who received new trials, instead of new sentence proceedings, because of the significant risk of injustice that the Government would suffer if it is forced to prosecute these defendants a second time. As we discussed above, see supra part III.A.2, the Government would spend substantial amounts of time and money retrying these defendants. Without any specific indication that the outcome in the trial court could have been tainted by bias, the public would most likely find it unjust to require the Government to suffer such costs. Thus, we conclude that there is an appreciable risk that vacating the defendants’ convictions would cause the public to lose confidence in our judicial system.16
As for the defendants who received only new sentencing hearings, we also conclude that the public’s confidence would be undermined if we required the Government to relitigate the trial court proceedings. Without evidence that bias could have tainted the outcome of the hearings, there is a significant risk that the public would find it unjust to require the Government to expend time and money to conduct these proceedings a second time. This risk is lower than the risk involved in granting the defendants a new trial, because the monetary and temporal costs to the Government to resentenee the defendants are lower. See supra part III.A.2. However, because a substantial risk remains that vacating the defendants’ sentences would undermine the public’s confidence, we conclude that factor three weighs against va-catur for the defendants who received new sentencing hearings.
IV.
In sum, the three factors identified by the Supreme Court in Liljeberg weigh strongly against vacatur in regard to the defendants who received new trials. First, the defendants have failed to show a risk of injustice to themselves while the Government has demonstrated a significant risk of injustice to it should the defendants’ convictions be vacated. Second, in light of the recently adopted protocol on the matter, there is little risk that failing to vacate the defendants’ convictions in these cases will produce injustice in other cases. Finally, public confidence in the judicial process would be harmed if these convictions — seemingly reached by fair procedures — were vacated, and the Government was required to retry the defendants.
*817Those same factors also weigh against vacatur, albeit less strongly, in regard to the cases in which Chief Judge O’Kelley granted only a new sentencing hearing. Although the risk of injustice to the Government is less in vacating the defendants’ sentences than in vacating the defendants’ convictions (because of the lesser burden of prosecuting a new sentencing proceeding as opposed to a new trial), that lesser risk is balanced against a complete absence of evidence regarding a risk of injustice to the defendants. Therefore, the first factor of the Liljeberg test still suggests that the remedy of vacatur is inappropriate. The second factor of Lilje-berg weighs against vacatur just as strongly in the sentencing context as in the context of the defendants’ convictions. Finally, public confidence in the judicial process is eroded (although perhaps to a lesser degree) by the setting aside of a fair sentence just as it is eroded by the setting aside of a fair conviction.
We therefore conclude that vacatur is an inappropriate remedy for the section 455(a) violation in these cases. Accordingly, the orders of the district court granting new trials and/or new sentencing hearings are
REVERSED.
. The facts of this case are set out more fully in the panel opinion. See United States v. Cerceda, 139 F.3d 847 (11th Cir.), vacated, *811161 F.3d 652 (11th Cir.1998) (en banc). We therefore present only a brief summary here.
. We refer to him as "Chief Judge," even though he no longer is, because that is the position Judge O’Kelley held at the time he was assigned these cases and entered some of the opinions and orders in them. See, e.g., United States v. Garrudo, 869 F.Supp. 1574, 1575 (S.D.Fla.1994).
. As a threshold matter, we must address the jurisdictional issue raised by the two appel-lees in United States v. Cerceda, No. 95-4628, and United States v. Hernandez, No. 95-4613. In those two cases. Chief Judge O’Kelley ordered a new trial as to sentencing only, and the appellees contend there is no statutory basis for an appeal of such orders. We disagree. Congress has provided the government may appeal from "a decision, judgment, or order of a district court ... granting a new trial after verdict or judgment.’’ 18 U.S.C. § 3731 (1994). That section provides that its provisions "shall be liberally construed to effectuate its purpose.” Id. The Supreme Court has said that "the purpose of the section was to remove all statutory barriers to Government appeals and to allow appeals whenever the Constitution would permit.” United States v. Wilson, 420 U.S. 332, 337, 95 S.Ct. 1013, 1019, 43 L.Ed.2d 232 (1975); accord, United States v. Posner, 764 F.2d 1535, 1538 (11th Cir.1985) ("it seems clear that [§ 3731] is designed only to prevent government appeals that would violate double jeopardy or interfere with an ongoing trial.”). Liberally construing § 3731 to effectuate its purpose, we hold that “a new trial after verdict or judgment” includes a new sentence proceeding.
Cerceda also argues that the government's notice of appeal in his case was untimely, because it was filed more than 30 days after the judgment, and under United States v. Rogers, 788 F.2d 1472, 1475 (11th Cir.1986), the government’s timely motion to reconsider did not stop the running of the Fed. R.App. P. 4(b) 30-day clock. ' That is correct, but we overrule that holding in Rogers for the reasons the Fifth Circuit gave in declining to adopt it. See United States v. Greenwood, 974 F.2d 1449, 1467—69 (5th Cir.1992). We hold that a timely motion to reconsider does toll the running of the 30-day period for filing a notice of appeal.
. Section 455(a) states that ”[a]ny justice, judge, or magistrate of the United States shall disqualify himself in any proceeding in which his impartiality might reasonably be questioned.” 28 U.S.C. § 455(a) (1994).
. The disposition of the defendants’ claims was divided into a number of opinions. Gar-rudo, which involved seven defendants who received new trials, is the only published district court opinion in this case.
. By affirming the district court’s holding on the violation issue, we are not adopting its reasoning. Cf. City Nat'l Bank of Miami v. General Coffee Corp., 828 F.2d 699, 703-04 (11th Cir.1987) (affirmance without opinion does not imply adoption of the trial court's reasoning). The opinion of a district court carries no precedential weight, even within the same district. See, e.g., State Farm Auto. Ins. Co. v. Bates, 542 F.Supp. 807, 816 (N.D.Ga.1982).
. Rule 60(b) allows a court to vacate a judgment on the basis of, inter alia, "newly discovered evidence which by due diligence could not have been discovered in time to move for a new trial under Rule 59(b)." Fed. R.Civ.P. 60(b)(2).
. In three of these cases. Chief Judge O'Kelley denied defendants’ motions for new trial but granted them new sentencing hearings, on the ground that Judge Moore became a subject of the investigation between the date they were convicted and the date they were sentenced. Notwithstanding representations to the contrary, there were no separate motions for new sentencing hearings based on the section 455(a) violation. (Of course, by denying the motions for new trial while at the same time granting new sentencing hearings, Chief Judge O’Kelley effectively treated the motions for new trial as motions for new sentencing hearings.)
.When a party requests the remedy of vaca-tur under Fed.R.Civ.P. 60(b) based on a violation of section 455(a), a court applying the first Liljeberg factor should consider as a threshold matter whether the motion was made within a reasonable time. See Fed. R.Civ.P. 60(b) (requiring party to file motion for relief "within a reasonable time, and [for certain types of motions] not more than one year after the judgment, order, or proceeding was entered or taken”); Liljeberg, 486 U.S. at 868, 108 S.Ct. at 2206-07 (stating that vaca-tur was appropriate "unless it can be said that respondent did not make a timely request *813for relief"); id. at 869, 108 S.Ct. at 2207 (noting that a delay of ten months between affirmance by the Court of Appeals on the merits and the filing of a motion to vacate "would normally foreclose relief based on a violation of § 455(a)”). The defendants in these cases brought their motions for new trial under Fed.R.Crim.P. 33 — a rule that, unlike Civil Rule 60(b), has very specific time limits within which a motion must be filed. It is therefore unclear whether the "reasonable time" requirement of Liljeberg applies in the criminal context. Because the Government does not contend that the motions were untimely, however, we do not consider the timeliness of each individual defendant's motion.
. In cases where the Court of Appeals reviews a district judge's challenged actions and affirms them on the merits either before or at the same time it considers whether the judge violated section 455(a), the possibility of a significant risk of injustice is substantially reduced — particularly if the review of the merits was plenary. See Parker, 855 F.2d at 1526 (concluding that judge’s potential bias presented no risk of injustice to party seeking vacatur because court exercised plenary review over merits in same appeal and concluded that district judge’s grant of summary judgment was proper); see also Air Line Pilots Ass’n, Int'l v. Continental Airlines, Inc. (In re Continental Airlines Corp.), 901 F.2d 1259, 1263 (5th Cir.1990) (reaching same conclusion where district court had previously affirmed challenged orders of bankruptcy court on merits); but see Liljeberg, 486 U.S. at 868 & n. 16, 108 S.Ct. at 2207 & n. 16 (recognizing that Court of Appeals panel affirmed on merits over a strong dissent and concluding that "a careful study of [the dissenting judge's] analysis of the merits of the underlying litigation suggests that there is a greater risk of unfairness in upholding the judgment in favor of Liljeberg than there is in allowing a new judge to take a fresh look at the issues”). All but one of the cases before us today, however, have not yet been subject to appellate review on the merits. The sole exception is United States v. Cerceda, 17 F.3d 1439 (11th Cir.1994), which we affirmed pursuant to 11th Cir. R. 36-1 on March 3, 1994.
. Liljeberg, 486 U.S. at 859, 108 S.Ct. at 2202, indicates that the seriousness of a section 455(a) violation could be mitigated by proof that the judge lacked knowledge of a disqualifying circumstance, and thus bear on the question of the appropriate remedy.
. Instead of making any showing of actual prejudice or the risk thereof, the defendants attempted to prevail on either an automatic vacatur or bias per se theory, or by focusing on the third Liljeberg factor alone. They were successful with that strategy in the district court, but not here. While the defendants got the district court to agree they could offer additional evidence on the violation issue if that court ruled against them on that issue, at no time have they been denied an opportunity to offer any evidence about, or otherwise make a showing on, the remedy issue. They never have requested such an opportunity.
. By contrast, the violation in Liljeberg was inexcusable. See Liljeberg, id. at 867, 108 S.Ct. at 2206.
. The protocol states:
Judicial officers who ... are informed that they are the subject or target of a federal criminal investigation for a crime that is punishable by imprisonment of one year or more may continue with their criminal and civil dockets and administrative duties until the Judicial Council determines to adopt limitations that the nature of the investigation and charges justify. However, after consultation with other judges of his or her court, the implicated judge may discontinue handling civil, criminal and administrative duties that the judge concludes the nature of the investigation and charges justify.
Eleventh Circuit Judicial Council, Protocol for Judicial Officers in the Eleventh Circuit in the Event of Arrest, Indictment or Possible Criminal Investigation (Sept. 5, 1996).
. This is especially true when the appellate court has conducted a plenary review of the trial court’s judgment — any risk that the trial court’s bias tainted the outcome in such a case has been cured by the de novo review of the unbiased court of appeals. Thus, the public would lose confidence in the judicial process if the trial court's judgment were vacated because that costly remedy would be unnecessary to ensure fairness to the parties. See O'Neill v. Continental Airlines, Inc. (In Matter of Continental Airlines), 981 F.2d 1450, 1463 (5th Cir.1993); Pfizer Inc. v. Kelly (In re School Asbestos Litig.), 977 F.2d 764, 787 (3rd Cir.1992).
. We note that this risk is mitigated by the Judicial Council’s protocol, discussed above, establishing guidelines for recusal in the event of a criminal investigation. The protocol reduces the likelihood that similar problems will arise in the future. Thus, although the public would most likely believe that vacating the defendants’ convictions is unjust, that belief would be mitigated by the understanding it is unlikely that vacating convictions for such reasons will be necessary again, and thus, that the Government will not have to retry fairly convicted defendants in the future.