Zeros of the derivative $\zeta'(s)$ of the Riemann zeta function

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Numbers

$n$

$\rho_n$

2.46316186945432128587439505331 + i * 23.2983204927628579020109616266

comment: The real and imaginary parts agree with OEIS A356216 and A356092.

1.28649682226904769704411427839 + i * 31.7082500831159086049543521423

2.30757006372263164161982156512 + i * 38.4899831730789358512452149000

1.38276360571167457578453372043 + i * 42.2909645545967298190807460934

0.964685622705685650525780173351 + i * 48.8471599050684790854189225685

2.10169990094877476831436781426 + i * 52.4321612451498357079460980394

1.89595976247123979784326762484 + i * 57.1347531990195337517383370900

0.848735328105403472052794047836 + i * 60.1408457820384239102072997439

1.20729562467416900725349107416 + i * 65.9199328242811617120903533822

10:

1.83294793165389012734216231451 + i * 68.6110788271283350210979210607

11:

1.77426908583776508787241516607 + i * 71.5281610651850349467736776682

12:

0.864622864426113300262052531442 + i * 76.3628078964670422358769577075

13:

1.3285155423330835459402056 + i * 78.6624059424066605569519626

14:

1.20356013488269005389299550659 + i * 83.6691335034148296303452016281

15:

2.39403928083969540187165954936 + i * 85.8020800349413089808844293334

16:

0.864103640598939499604405698998 + i * 88.1775174098810128722273933318

17:

1.30408781494307691801197996837 + i * 93.0859268156198809610928605804

18:

0.780628004724644645328178951938 + i * 95.2929682713522169397105676541

19:

1.79843738976540742229326672130 + i * 98.8269718674541577083227561190

20:

1.76710606124689989874566437400 + i * 101.715421432538018621640749363

21:

1.16055253054723397972155233137 + i * 104.503209175654916233158432775

22:

1.09246390774683122554463202013 + i * 106.560732444777412706696849714

23:

0.635638410195870078269381432276 + i * 111.431017613746736506374036932

24:

1.90599058734168480385542257767 + i * 113.630627302549582735957930953

25:

1.44733504242649614597372103797 + i * 115.583388477428138757970502010

26:

1.65708653073366653438423716540 + i * 118.908491963851061312041014484

27:

1.02259521669029785707299335382 + i * 121.999396480142649712191295949

28:

0.847766864212029839013392894815 + i * 123.715269749934938660387116189

29:

1.37922767858206723878450447413 + i * 127.964537384854636220896181499

30:

1.05704331510860777489243627077 + i * 130.322213224776924941828204985

31:

2.32846049714225134228608622987 + i * 132.485520308364405480820274306

32:

0.856309339180055369726489850143 + i * 134.193836602386409228845589451

33:

1.04332789131225610212908949841 + i * 138.659121570893262937453010016

34:

0.943828539659771481951278713963 + i * 140.469959838197100688840071351

35:

1.30556394063897816655016913264 + i * 142.650359963440026363339146050

36:

1.01880715023523717679694592744 + i * 146.630738119460117999518837832

37:

2.42377569992720686004396088242 + i * 147.874503354183342898857850814

38:

0.662929906884329935358394515817 + i * 150.485953620246866996387671328

39:

1.26598676313464538711175653874 + i * 152.613294885666000215081429163

40:

0.966951342073371224843720646498 + i * 156.632667913413661808966416348

41:

0.863404697829980428874709939776 + i * 158.282522106715305651648590786

42:

1.77583103312831478427628530186 + i * 161.021873315306938337143947060

43:

1.55034564239770281980837028395 + i * 162.665563695696057103327890460

44:

1.20102789459661819186979479511 + i * 166.087962357276920402326320269

45:

1.41410183563062734723879957649 + i * 167.894407526578422294488757989

46:

0.645462688860882537571384288692 + i * 169.537848080519584802511674409

47:

0.928844920181084287595025255923 + i * 173.927146629227684060743832297

48:

1.34419459217255776181353425284 + i * 175.666781393198914078405295217

49:

1.82217964533340480022554604803 + i * 177.608899654623140363340214668

50:

1.16999263062720348364780686408 + i * 179.331429604995658665497951955

51:

1.44656430851625910301991792117 + i * 182.221596359705832709323414939

52:

0.615980865250224349344163629535 + i * 185.214812338050414602642586851

53:

1.04987712040973067471425509172 + i * 186.713092049386239602612706685

54:

1.39820225661929369615906605475 + i * 189.239599101036758172475041008

55:

0.773908639802288184756721916796 + i * 192.517146328426982392104624921

56:

2.30538305740399481221348360291 + i * 194.119329189410674857436467804

57:

1.24213485292606016540435621348 + i * 195.912464479535423146272472425

58:

0.810904075519309672986777587512 + i * 197.545544880285132110535613614

Definition

$\zeta(s)$ is the Riemann zeta function [2]. The table holds the non-real zeros $\rho_n$ of its derivative $\zeta'(s)$ with $\operatorname{Im}\rho_n>0$, ordered by increasing imaginary part and indexed from $n=1$.

Parameters

$n$

— index of the zero ($n\geq1$)

Formulas

(1)

$\zeta'(\overline{s})=\overline{\zeta'(s)}$.

Comments

(2)

Since $\zeta'(\overline{s})=\overline{\zeta'(s)}$, the conjugate $\overline{\rho_n}$ is a zero whenever $\rho_n$ is. Only the zeros with positive imaginary part are entries.

(3)

The real zeros of $\zeta'$ are not entries. There is one such zero in each interval $(-2m-2,-2m)$ for $m\geq1$; the first is $-2.717262829204574\ldots$ [4].

(4)

Speiser's theorem, in the form used by Levinson and Montgomery [1], says that the Riemann hypothesis is equivalent to $\zeta'$ having no non-real zeros with $0<\operatorname{Re}s<1/2$.

Programs

(P1)

Sage

import numberdb.sage
from sage.rings.complex_arb import ComplexBallField

C = ComplexBallField(200)
z = C("2.4631618694543212858743950533",
      "23.2983204927628579020109616")
for _ in range(5):
    z = z - z.zetaderiv(1) / z.zetaderiv(2)
print(z)

References

[1]

N. Levinson and H. L. Montgomery, Zeros of the derivatives of the Riemann zeta-function, Acta Math. 133 (1974), 49-65. (doi)

Links

[2]

DLMF 25.2: Definition and Expansions

[3]

OEIS A356216: real part of the first zero of $\zeta'(s)$

[4]

OEIS A356092: imaginary part of the first zero of $\zeta'(s)$

Similar tables

Zeros of the Riemann zeta function — stores the imaginary parts of zeros of $\zeta(s)$ itself; Speiser's theorem relates the Riemann hypothesis to the horizontal distribution of the zeros stored here

Values of the derivative $\zeta'(s)$ of the Riemann zeta function at rational numbers — stores values of the same function $\zeta'(s)$ at rational arguments rather than its zeros

Values of the logarithmic derivative $\zeta'(s)/\zeta(s)$ of the Riemann zeta function at rational numbers — stores the logarithmic derivative; away from zeros of $\zeta(s)$, a zero of $\zeta'(s)$ is a zero of $\zeta'(s)/\zeta(s)$

Zeros of Dirichlet $L$-functions — stores zero ordinates for Dirichlet $L$-functions

Zeros of the $L$-functions of level one cusp forms — stores zero ordinates for another family of $L$-functions

Zeros of the Dedekind zeta functions of cubic fields — stores zero ordinates for Dedekind zeta functions in the same proposal family

Keiper-Li coefficients — stores the coefficients whose positivity is Li's criterion for the Riemann hypothesis

Taylor coefficients of the completed Riemann zeta function at $1/2$ — stores coefficients at the centre of the critical strip for another Riemann-hypothesis criterion

Data properties

Entries are of type: complex number

How well the digits are known: proven

How they were obtained:

The generator locates candidates by Newton iteration applied to $f(s)=\zeta'(s)$ and $f'(s)=\zeta''(s)$ in arb complex ball arithmetic. It runs two seed grids, one on half-integer heights and one on offset quarter-integer heights, and refuses to continue unless both grids find the same 58 candidates with $0<\operatorname{Im}\rho\leq200$.

Each candidate is refined and then certified by a Krawczyk step on a complex ball $B$: with $m$ the centre of $B$ and $Y=1/\zeta''(m)$, the computed set $m-Y\zeta'(m)+(1-Y\zeta''(B))(B-m)$ must lie in the interior of $B$. This proves that $B$ contains a unique zero of $\zeta'$. The first entry is checked against the OEIS decimal expansions [3] and [4].

Table is complete: no (it holds the 58 certified non-real zeros found by the generator with $0<\operatorname{Im}\rho\leq200$; both seed grids found the same set)