Values of the logarithmic derivative $\zeta'(s)/\zeta(s)$ of the Riemann zeta function at rational numbers

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Numbers

$s$

$\zeta'(s)/\zeta(s)$

1.837877066409345483560659472811235279722794947275566825634303080965531391854520795389486597271908395

comment: $\log(2\pi)$.

1/2:

2.686091709612832791116478748724871144507269625811776921584451315495547282857339698426088863613775527

comment: $\frac12(\frac\pi2+\gamma+\log(8\pi))$.

-1:

1.985053724405411150567035922913367713168436564022421403998267676288316395998926073114301203990105066

-0.5699609930945328063998643600197300024034822808069309795581973604379172773667406067878675434264499217

comment: $\gamma+\log(2\pi)-12\log A$, where $A$ is the Glaisher-Kinkelin constant.

-0.1648226821582772401864933947994750650230095804224032708717453372762869598655773519080862678086888914

-3:

0.6454291629329161373300369052496611474797327690754608471336427867989402397508883503314124644990819921

-0.06366976495537112649619867568935677004811181919330375602690580428187445445203621733831213726876018063

-0.02755619219153047054461103190802827770376028295880322922674541929458396936254459150728987967871724563

-5:

0.1443924670100560716577305255061363773400047973822793468908395540114292279004094769820806342811965275

-0.01263306903251106082389229594583199990838384750012225578410257149436344260155734398898030705087471609

-0.005983558570638396153781990333460225134012328034877668923220680346408402770881248774204870032476375934

-7:

-0.1748742432382177565921360051160865468566930530221003964727326769942234109265543791255791460033078511

-0.002890168308046756383549574847418599521209806619552036230054150012520327584117297405130050575894146993

-0.001413144078811703121012095454036393626895725107635591686008221013599789501139270525809732170719772623

-9:

-0.4131791822120916036502014143836789647523562942435601073674775831173825517887129412488556733896021093

10:

-0.0006963404452840204365952766909372927366576765092034364464203550004722978330698463929646343007109998839

11:

-0.0003448518005384273275108214836032865189778239275820408156661224266380105051939475265309041019086924813

-11:

-0.6046132728852498614746713965421955826315305032617682082586844744124014587895109480758593638906759630

12:

-0.0001713406812166717030343854415115839483925584000862446461225546145442999231809304750518528905462371316

13:

-0.00008531343955816884070010228640739709656841860478824819474155623821058274564192234646014170985102771613

-13:

-0.7649984849349225634324641776800206677859282637531018966212106306528447704188002451365112137842991583

14:

-0.00004253888795422615549801456009675520425120816516281269385280863051124470414804367081025940717945212122

15:

-0.00002123043613140305293086512223345424732880022557423978622797904006874646160679029754140368306765212525

-15:

-0.9031256596381091638826694473225917048278385667493671068173383057601569095933725071153921891226913662

16:

-0.00001060228000572094338798301276381340043614326413569773582037161843720076767101978716737930688248227140

17:

-0.000005296842904493593945793228089119436323902472655806950795822860404533825328648379360515749480038557139

-17:

-1.024457144337005122901759356892389267438464780134282057609368575253069414067887142512580755958424273

18:

-0.000002646992874467806651014619436839025104047526279570473201866831407049234332854978214106588796634339361

19:

-0.000001323021170172668765606229135268704671782437647317549620927532979086906931639368944674266882507270590

-19:

-1.132646264480413735676734632766605378965782753206629221783394820544857005451699333016805160959280492

20:

-6.613523898316398628732481447041059732089568563829073627560425434344587077989769136781961275493643943e-7

Definition

$\zeta(s)$ is the Riemann zeta function, $\sum_{n\geq1}n^{-s}$ for $\Re(s)>1$ continued meromorphically away from its pole at $s=1$ [1]. The table holds its logarithmic derivative $\zeta'(s)/\zeta(s)$ at rational $s$ where it is finite.

Parameters

$s$

— argument ($s\in\mathbb{Q}$; $s\neq1$; $\zeta(s)\neq0$)

Formulas

(1)

For $\Re(s)>1$, $-\zeta'(s)/\zeta(s)=\sum_{n\geq1}\Lambda(n)n^{-s}$ [3].

(2)

For noninteger $s$ where both logarithmic derivatives are finite, $\frac{\zeta'(1-s)}{\zeta(1-s)}=\log(2\pi)+\frac\pi2\tan\frac{\pi s}2-\psi(s)-\frac{\zeta'(s)}{\zeta(s)}$ [2].

Comments

(3)

$\Lambda(n)$ is the von Mangoldt function, equal to $\log p$ when $n$ is a positive power of a prime $p$ and to $0$ otherwise. The table holds $\zeta'(s)/\zeta(s)$ itself, not its negative, which is the Dirichlet series in (1).

(4)

$\zeta'(s)/\zeta(s)$ has simple poles at $s=1$ and at the zeros of $\zeta(s)$. The negative even trivial zeros $s=-2,-4,\ldots$ are therefore not listed.

(5)

The integer orders are the logarithmic forms of the integer zeta special values and derivatives [4]; the nonpositive even zeros are excluded. At the functional-equation centre, $\zeta'(1/2)/\zeta(1/2)=(\log\pi-\psi(1/4))/2$ [5]. The integer interval is a finite reference prefix; denominator bounds alone do not select additional orders.

Programs

(P1)

Python

from values import SelectedValues
from numberdb._write import to_text

generator = SelectedValues('T246')
for params in generator.enumerate():
    print(params, to_text(generator.value(params, 100), 100))

Links

[1]

DLMF 25.2: Definition and Expansions

[2]

DLMF 25.4: Reflection Formulas

[3]

DLMF 27.4: Euler Products and Dirichlet Series

[4]

DLMF 25.6: Integer Arguments and Derivatives

[5]

DLMF 25.4: Reflection Formulas

Similar tables

Values of the derivative $\zeta'(s)$ of the Riemann zeta function at rational numbers — stores $\zeta'(s)$; dividing those values by the corresponding values of $\zeta(s)$ gives this table where $\zeta(s)\neq0$, but a reader holding the quotient is likely looking for the logarithmic derivative itself

Values of the Riemann zeta function at rational numbers — stores $\zeta(s)$ at the same kind of rational arguments, the denominator in $\zeta'(s)/\zeta(s)$

Laurent series coefficients of the logarithmic derivative of the Riemann zeta function at 1 — stores the Laurent expansion of this same logarithmic derivative at its pole at $s=1$

Values of the Hurwitz zeta function at pairs of rational numbers — $\zeta(s)$ is the Hurwitz zeta function at $x=1$

Data properties

Entries are of type: real number

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 478 relative accuracy bits. All 31 entries also pass mpmath checks at 150 and 200 decimal digits; 2 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.

Table is complete: no (it holds the integer arguments $-20\leq s\leq20$ where the quotient is finite, and $s=1/2$; these are the integer special-value references and the functional-equation centre, not a continuous rational grid)