A058303 -id:A058303 - cp's OEIS frontend

A305742 Decimal expansion of imaginary part of 7th nontrivial zero of Riemann zeta function.

4, 0, 9, 1, 8, 7, 1, 9, 0, 1, 2, 1, 4, 7, 4, 9, 5, 1, 8, 7, 3, 9, 8, 1, 2, 6, 9, 1, 4, 6, 3, 3, 2, 5, 4, 3, 9, 5, 7, 2, 6, 1, 6, 5, 9, 6, 2, 7, 7, 7, 2, 7, 9, 5, 3, 6, 1, 6, 1, 3, 0, 3, 6, 6, 7, 2, 5, 3, 2, 8, 0, 5, 2, 8, 7, 2, 0, 0, 7, 1, 2, 8, 2, 9, 9, 6, 0, 0, 3, 7, 1, 9

Offset: 2

Seiichi Manyama, Jun 23 2018

			The zero is at 1/2 + i * 40.918719012147495187398126914633254395726165962777...

Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1), A065434 (k=2), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), this sequence (k=7), A305743 (k=8), A305744 (k=9), A306004 (k=10).

Cf. A002410 (rounded values: main entry), A013629 (floor), A092783 (ceiling).

RealDigits[Im[ZetaZero[7]], 10, 120][[1]] (* Vaclav Kotesovec, Jun 23 2018 *)

lfunzeros(1,[40,41])[1] \\ M. F. Hasler, Nov 23 2018

A305743 Decimal expansion of imaginary part of 8th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

4, 3, 3, 2, 7, 0, 7, 3, 2, 8, 0, 9, 1, 4, 9, 9, 9, 5, 1, 9, 4, 9, 6, 1, 2, 2, 1, 6, 5, 4, 0, 6, 8, 0, 5, 7, 8, 2, 6, 4, 5, 6, 6, 8, 3, 7, 1, 8, 3, 6, 8, 7, 1, 4, 4, 6, 8, 7, 8, 8, 9, 3, 6, 8, 5, 5, 2, 1, 0, 8, 8, 3, 2, 2, 3, 0, 5, 0, 5, 3, 6, 2, 6, 4, 5, 6, 3, 4, 9, 3, 7, 1

Offset: 2

Seiichi Manyama, Jun 23 2018

			The zero is at 1/2 + I*43.3270732809149995194961221654068... - _M. F. Hasler_, Nov 21 2018

Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function
OEIS index entries for sequences related to the zeta function.

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1), A065434 (k=2), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), this sequence (k=8), A305744 (k=9), A306004 (k=10).

Cf. A002410 (rounded values: main entry), A013629 (floor), A092783 (ceiling).

RealDigits[Im[ZetaZero[8]], 10, 120][[1]] (* Vaclav Kotesovec, Jun 23 2018 *)

solve(X=43,44,imag(zeta(0.5+X*I))) \\ M. F. Hasler, Nov 21 2018

lfunzeros(1,[43,44])[1] \\ M. F. Hasler, Nov 23 2018

A305744 Decimal expansion of imaginary part of 9th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

4, 8, 0, 0, 5, 1, 5, 0, 8, 8, 1, 1, 6, 7, 1, 5, 9, 7, 2, 7, 9, 4, 2, 4, 7, 2, 7, 4, 9, 4, 2, 7, 5, 1, 6, 0, 4, 1, 6, 8, 6, 8, 4, 4, 0, 0, 1, 1, 4, 4, 4, 2, 5, 1, 1, 7, 7, 7, 5, 3, 1, 2, 5, 1, 9, 8, 1, 4, 0, 9, 0, 2, 1, 6, 4, 1, 6, 3, 0, 8, 2, 8, 1, 3, 3, 0, 3, 3, 5, 3, 7, 2, 3

Offset: 2

Seiichi Manyama, Jun 23 2018

			The zero is at 1/2 + i * 48.0051508811671597279424727494275160416868440011444251...

Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function
OEIS index entries for sequences related to the zeta function.

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1), A065434 (k=2), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), this sequence (k=9), A306004 (k=10).

Cf. A002410 (rounded values: main entry), A013629 (floor), A092783 (ceiling).

RealDigits[Im[ZetaZero[9]], 10, 120][[1]] (* Vaclav Kotesovec, Jun 23 2018 *)

lfunzeros(1,[48,49])[1] \\ M. F. Hasler, Nov 23 2018

Edited (example, link, cross-references) by M. F. Hasler, Nov 23 2018

A306004 Decimal expansion of imaginary part of 10th nontrivial zero of Riemann zeta function.

Original entry on oeis.org

4, 9, 7, 7, 3, 8, 3, 2, 4, 7, 7, 6, 7, 2, 3, 0, 2, 1, 8, 1, 9, 1, 6, 7, 8, 4, 6, 7, 8, 5, 6, 3, 7, 2, 4, 0, 5, 7, 7, 2, 3, 1, 7, 8, 2, 9, 9, 6, 7, 6, 6, 6, 2, 1, 0, 0, 7, 8, 1, 9, 5, 5, 7, 5, 0, 4, 3, 3, 5, 1, 1, 6, 1, 1, 5, 1, 5, 7, 3, 9, 2, 7, 8, 7, 3, 2, 7, 0, 7, 5, 0

Offset: 2

Seiichi Manyama, Jun 23 2018

			The zero is at 1/2 + i * 49.77383247767230218191678467856372405772317829967666...

Andrew M. Odlyzko, Tables of zeros of the Riemann zeta function
OEIS index entries for sequences related to the zeta function.

Imaginary part of k-th nontrivial zero of Riemann zeta function: A058303 (k=1), A065434 (k=2), A065452 (k=3), A065453 (k=4), A192492 (k=5), A305741 (k=6), A305742 (k=7), A305743 (k=8), A305744 (k=9), this sequence (k=10).

Cf. A002410 (rounded values: main entry), A013629 (floor), A092783 (ceiling).

RealDigits[Im[ZetaZero[10]], 10, 120][[1]] (* Vaclav Kotesovec, Jun 23 2018 *)

lfunzeros(1,[49,50])[1] \\ M. F. Hasler, Nov 23 2018

Edited (added link, example, cross-reference) by M. F. Hasler, Nov 23 2018

A254297 Consider the nontrivial zeros of the Riemann zeta function on the critical line 1/2 + i*t and the gap, or first difference, between two consecutive such zeros; a(n) is the lesser of the two zeros at a place where the gap attains a new minimum.

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 14, 20, 25, 28, 35, 64, 72, 92, 136, 160, 187, 213, 299, 316, 364, 454, 694, 923, 1497, 3778, 4766, 6710, 18860, 44556, 73998, 82553, 87762, 95249, 354770, 415588, 420892, 1115579, 8546951

Offset: 1

Robert G. Wilson v, Jan 27 2015

nonn

Since all zeros are assumed to be on the critical line, the gap, or first difference, between two consecutive zeros is measured as the difference between the two imaginary parts.
Inspired by A002410.
No other terms < 10000000. The minimum gap so far is 0.002323...

			a(1)=1 since the first Riemann zeta zero, 1/2 + i*14.13472514... (A058303) has no previous zero, so its gap is measured from 0.
a(2)=2 since the second Riemann zeta zero, 1/2 + i*21.02203964... (A065434) has a gap of 6.887314497... which is less than the previous gap of ~14.13472514.
a(3)=3 since the third Riemann zeta zero, 1/2 + i*25.01085758... (A065452) has a gap of 3.988817941... which is less than ~6.887314497.
The fourth Riemann zeta zero, 1/2 + i*30.42487613... (A065453) has a gap of 5.414018546... which is not less than ~6.887314497 and therefore is not in the sequence.
a(4)=5 since the fifth Riemann zeta zero, 1/2 + i*32.93506159... (A192492) has a gap of 2.510185462... which is less than ~3.988817941.
a(5)=8 since the eighth Riemann zeta zero, 1/2 + i*43.32707328...  has a gap of 2.408354269... which is less than ~2.510185462.

Glen Pugh, The Riemann Hypothesis in a Nutshell.

Cf. A002410, A100060, A161914, A117538, A153595, A326502.

k = 1; mn = Infinity; y = 0; lst = {}; While[k < 10001, z = N[ Im@ ZetaZero@ k, 64]; If[z - y < mn, mn = z - y; AppendTo[lst, k]]; y = z; k++]; lst

a(n) = A326502(n) + 1. - Artur Jasinski, Oct 24 2019

a(38) from Arkadiusz Wesolowski, Nov 08 2015

a(39) from Artur Jasinski, Oct 24 2019

A123504 Sequence generated from the first nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0

Offset: 2

Gary W. Adamson, Oct 01 2006

nonn

A123505 records the lengths of runs. A123506 uses the second zero.

			a(8) = 1 since (1/8)^z = (0.353553..., angle 115.943... degrees).

Cf. A058303, A100060, A102522, A102523, A123505, A123506, A123507.

a[n_] := Boole[Arg[1/n^ZetaZero[1]] > 0]; Array[a, 100, 2] (* Amiram Eldar, May 31 2025 *)

t=1/2+solve(y=14,15,imag(zeta(1/2+y*I)))*I;
a(n)=arg(n^-t)>0 \\ Charles R Greathouse IV, Mar 10 2016

Extract argument from (1/n)^z, z = (1/2 + i*14.1347251417...). a(n) = 1 if the argument is between 0 and 180 degrees, and = 0 if otherwise (n = 2, 3, 4, ...).

More terms from Amiram Eldar, May 31 2025

A199499 Imaginary part of first zeta zero divided by 2*Pi.

Original entry on oeis.org

2, 2, 4, 9, 6, 1, 1, 3, 7, 5, 5, 5, 2, 3, 6, 7, 4, 2, 4, 2, 4, 3, 2, 7, 0, 7, 1, 1, 5, 9, 0, 0, 7, 8, 6, 9, 5, 0, 5, 9, 9, 9, 2, 7, 5, 2, 9, 0, 8, 8, 8, 9, 2, 8, 6, 1, 2, 5, 0, 2, 9, 4, 7, 7, 4, 6, 4, 4, 7, 0, 2, 9, 1, 5, 0, 7, 3, 3, 8, 6, 3, 9, 7, 6, 4, 0, 6, 1, 0, 0

Offset: 1

Mats Granvik, Nov 07 2011

2.249611375...= Im(ZetaZero(1))/(2*Pi);

2.249612177...= Re(zeta(1/ZetaZero(2))*(1-1/(11/2)^(1/ZetaZero(2)-1))). - Mats Granvik, Mar 10 2016

			2.249611375552367424243270711590078695...

Cf. A058303.

RealDigits[N[Im[ZetaZero[1]]/(2*Pi), 90]]

solve(y=14,15,imag(zeta(1/2+y*I)))/2/Pi \\ Charles R Greathouse IV, Mar 10 2016

lfunzeros(lzeta,[14,15])[1]/2/Pi \\ Charles R Greathouse IV, Mar 10 2016

A373204 Decimal expansion of the imaginary part of the first zero, for real(s) >= 1/2, of the function Psi(s) = Sum_{n>=1} 1/n!^s.

Original entry on oeis.org

4, 9, 0, 6, 8, 7, 6, 4, 3, 5, 1, 4, 2, 8, 5, 1, 3, 4, 7, 5, 3, 5, 1, 0, 8, 2, 5, 8, 3, 5, 5, 8, 5, 3, 5, 3, 1, 5, 3, 2, 8, 5, 6, 4, 6, 4, 8, 9, 9, 3, 3, 7, 6, 3, 5, 2, 0, 2, 8, 8, 9, 5, 2, 4, 8, 7, 0, 0, 8, 0, 9, 6, 8, 4, 9, 1, 6, 0, 4, 0, 6, 0, 1, 1

Offset: 1

Roberto Trocchi, Jun 21 2024

Defining the Psi function to be Psi(s) = Sum_{n>=1} 1/n!^s, the first zero, for real(s) >= 1/2, is approximately s1 = 0.6418158643 + 4.9068764351*i.
All the zeros of the Psi function seem (conjecturally) to be in the critical strip 0 < real(s) <= 1.
Moreover, all the zeros of the Psi function seem (conjecturally) to be in the strip 0 < real(s) <= 0.73. [There is obviously something wrong here! - N. J. A. Sloane, Dec 30 2024]
See my document on the zeros of the Psi function on the complex plane.

			4.9068764351428513475351082583558535315328564648993...

Roberto Trocchi, The Psi function and its zeros on the complex plane, June 21 2024.

Cf. A091131, A058303.

Psi[s_, nmax_] := ParallelSum[1/n!^s, {n, 1, nmax}]
FindRoot[{Re[Psi[x + y*I, 2000]], Im[Psi[x + y*I, 2000]]}, {{x, 1/2}, {y, 5}}, WorkingPrecision -> 1000][[2]][[2]]

Imaginary part of the first zero for real(s) >= 1/2, Psi(s) = 0, where Psi(s) = Sum_{n>=1} 1/n!^s.

A131583 Concatenation of first n numbers of the decimal expansion of imaginary part of first nontrivial zero of Riemann zeta function.

Original entry on oeis.org

1, 14, 141, 1413, 14134, 141347, 1413472, 14134725, 141347251, 1413472514, 14134725141, 141347251417, 1413472514173, 14134725141734, 141347251417346, 1413472514173469, 14134725141734693, 141347251417346937

Offset: 0

Omar E. Pol, Sep 13 2007

Index entries for zeta function.

See A002410 and A058303 for more information.

Module[{nn=20,zz},zz=RealDigits[Im[ZetaZero[1]],10,nn][[1]];Table[ FromDigits[ Take[zz,n]],{n,nn}]] (* Harvey P. Dale, May 23 2019 *)

A133010 Characteristic function of the Riemann zeta function: If n is a nearest integer to imaginary part of zero, then a(n)=1 else a(n)=0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0

Offset: 1

Omar E. Pol, Sep 13 2007

Also an interesting triangle read by row: See tabl.

			a(30)=1 because 30 is the nearest integer to imaginary part of 4th nontrivial zero of Riemann zeta function.

See A002410 for more information. Cf. A058303, A065434, A065452, A065453.

cp's OEIS Frontend

A305742 Decimal expansion of imaginary part of 7th nontrivial zero of Riemann zeta function.

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A305743 Decimal expansion of imaginary part of 8th nontrivial zero of Riemann zeta function.

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A305744 Decimal expansion of imaginary part of 9th nontrivial zero of Riemann zeta function.

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A306004 Decimal expansion of imaginary part of 10th nontrivial zero of Riemann zeta function.

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A254297 Consider the nontrivial zeros of the Riemann zeta function on the critical line 1/2 + i*t and the gap, or first difference, between two consecutive such zeros; a(n) is the lesser of the two zeros at a place where the gap attains a new minimum.

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A123504 Sequence generated from the first nontrivial zero of the Riemann zeta function.

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A199499 Imaginary part of first zeta zero divided by 2*Pi.

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A373204 Decimal expansion of the imaginary part of the first zero, for real(s) >= 1/2, of the function Psi(s) = Sum_{n>=1} 1/n!^s.