Values of the Riemann zeta function at rational numbers

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Numbers

$s$

$\zeta(s)$

-1/2

1/2:

-1.460354508809586812889499152515298012467229331012581490542886087825530529474500625276419375463356819

-1:

-1/12

3/2:

2.612375348685488343348567567924071630570800652400063407573328248814927767688272860996243868126311952

1.644934066848226436472415166646025189218949901206798437735558229370007470403200873833628900619758705

-2:

5/2:

1.341487257250917179756769693348612136623037629505986511253796728340918923813185441581761085998697994

1.202056903159594285399738161511449990764986292340498881792271555341838205786313090186455873609335258

-3:

1/120

1.082323233711138191516003696541167902774750951918726907682976215444120616186968846556909635941699917

-4:

1.036927755143369926331365486457034168057080919501912811974192677903803589786281484560043106557133336

-5:

-1/252

1.017343061984449139714517929790920527901817490032853561842408664004332182901957897882773977938535170

-6:

1.008349277381922826839797549849796759599863560565238706417283136571601478317355735346096968913851323

-7:

1/240

1.004077356197944339378685238508652465258960790649850020329110202652582952574748814395287230372371971

-8:

1.002008392826082214417852769232412060485605851394888756548596615909785053390258398950393069127169586

-9:

-1/132

10:

1.000994575127818085337145958900319017006019531564477517257788994636291465151912954397041968610385652

-10:

11:

1.000494188604119464558702282526469936468606435758208617119141436100054059798219814702591843023560629

-11:

691/32760

12:

1.000246086553308048298637998047739670960416088458003404533040952133252019681940913049042808551900699

-12:

13:

1.000122713347578489146751836526357395714275105895509845136702671620896726829844209812892713953268135

-13:

-1/12

14:

1.000061248135058704829258545105135333747481696169154549482755202252862941023177420876659782971998467

-14:

15:

1.000030588236307020493551728510645062587627948706858177506569932893332267156342279573072334347017548

-15:

3617/8160

16:

1.000015282259408651871732571487636722023237388990471531153105203588787087027953151786285604846322462

-16:

17:

1.000007637197637899762273600293563029213088249090262679095379843972935643290282459342081738636916671

-17:

-43867/14364

18:

1.000003817293264999839856461644621939730454697218953331143174429987630039542650045638001968668989649

-18:

19:

1.000001908212716553938925656957795101353258571144838630235933046761823949705341309312664227118076302

-19:

174611/6600

20:

1.000000953962033872796113152038683449345943794187410595750056489851137513731143900257836097976387478

-20:

21:

1.000000476932986787806463116719604373045966446694784937600207487376596839087898159833876638564497256

-21:

-77683/276

22:

1.000000238450502727732990003648186752994935041821779658269849603116474458935622914821316156167743985

-22:

23:

1.000000119219925965311073067788718882326387254997784519858603225797236243730427435123174313352235859

-23:

236364091/65520

24:

1.000000059608189051259479612440207935801227503918837302795864246972321724495355468544848206832825003

-24:

25:

1.000000029803503514652280186063705069366011844730919543312398681339013384460767464082069171562896200

-25:

-657931/12

26:

1.000000014901554828365041234658506630698628864788167885910547435968789971296744862510258486179405654

-26:

27:

1.000000007450711789835429491981004170604119454719031882565829993239578352147606271570867900837100313

-27:

3392780147/3480

28:

1.000000003725334024788457054819204018402423232893059295811519769334706169604960304364973738801930066

-28:

29:

1.000000001862659723513049006403909945416948061665330469200665774893805558091693265817877381474521004

-29:

-1723168255201/85932

30:

1.000000000931327432419668182871764735021219813567955136816185008613360441960672940496363503624604027

-30:

31:

1.000000000465662906503378407298923325122007106269185336947307372971693371175669889809582649582191406

-31:

7709321041217/16320

32:

1.000000000232831183367650549200145597594049502482982284530311077602258387912189392170586790714721837

-32:

33:

1.000000000116415501727005197759297383545630951652247172763593256517739947029124624567548673934974376

-33:

-151628697551/12

34:

1.000000000058207720879027008892436859891063054173122604617215955071688124163071396179208265960455374

-34:

35:

1.000000000029103850444970996869294252278840464106981987433032256210254825640488901408204338569117244

-35:

26315271553053477373/69090840

36:

1.000000000014551921891041984235929632245318420983808894124038069139542218571745865030220152998942329

-36:

37:

1.000000000007275959835057481014520869012338059264850925555466107705796942638438370439334710114183793

-37:

-154210205991661/12

38:

1.000000000003637979547378651190237236355873273512646028384897469947951594042714249816909036997016960

-38:

39:

1.000000000001818989650307065947584832100730085030589309618664070535251253356509316152407149980168289

-39:

261082718496449122051/541200

40:

1.000000000000909494784026388928253311838694908753860000990878828505479710112025368695607103530607220

-40:

Definition

The list contains values $\zeta(s)$ of the Riemann zeta function at rational numbers $s$.

Parameters

$s$

— rational number ($s \neq 1$)

Formulas

(1)

$\zeta(-n) = (-1)^n\frac{B_{n+1}}{n+1}$ for integers $n\geq 0$, where $B_n$ is the $n$'th Bernoulli number with sign convention $B_1=-1/2$.

(2)

$\zeta(2n) = \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}$ for positive integers $n$.

Comments

(3)

The Riemann zeta function $\zeta(s)$ is the meromorphic continuation of the Dirichlet series $\sum_{n=1}^\infty n^{-s}, \Re(s) > 1$.

(4)

At $s=1$, $\zeta$ has a simple pole.

(5)

Integer arguments belong to the Bernoulli-number, even-zeta and odd-zeta special-value families [2]. The point $s=1/2$ is fixed by the functional equation $s\mapsto1-s$ [3]. The orders $3/2$ and $5/2$ occur in the particle-number and energy integrals of the three-dimensional ideal Bose gas at unit fugacity [4]. These reasons do not single out the other nonintegral rational orders of a denominator-bounded mesh.

Programs

(P1)

Python

from values import SelectedValues
from numberdb._write import to_text

generator = SelectedValues('T14')
for params in generator.enumerate():
    print(params, to_text(generator.value(params, 100), 100))

Links

[1]

Wikipedia: Riemann zeta function

[2]

DLMF 25.6: Integer Arguments

[3]

DLMF 25.4: Reflection Formulas

[4]

David Tong: Statistical Physics, Chapter 3, Bose gases

Data properties

Entries are of type: real number

Table is complete: no (it holds the integer arguments $-40\leq s\leq40$, $s\ne1$, the functional-equation centre $s=1/2$, and the Bose-gas orders $s=3/2,5/2$; the integer cutoff is a finite reference prefix, not a completeness claim beyond that range)

How well the digits are known: proven

How they were obtained:

Every selected entry is checked against a fresh exact rational result or an Arb enclosure at 493 working bits (100 decimal digits plus 160 guard bits). Rational arguments and square roots are enclosed before evaluation, so their uncertainty is propagated.

Existing decimal strings and surviving row annotations are retained only after the recomputed enclosure is strictly contained in the interval their last digit denotes. New inexact entries are written to 100 significant digits; proved rational values remain exact. Every inexact ball is finite and has at least 491 relative accuracy bits. All 83 entries also pass mpmath checks at 150 and 200 decimal digits; 20 entries have additional closed-form or functional-identity checks. The mpmath agreements supplement, and do not establish, the rigorous enclosure. A nonzero control and ten-last-place perturbation controls pass. The attached checks-evidence.json records this selection, and checks.py reproduces the checks without writing to the website.