A259072 -id:A259072 - cp's OEIS frontend

A266273 Decimal expansion of zeta'(-18) (the derivative of Riemann's zeta function at -18) (negated).

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

Offset: 2

G. C. Greubel, Dec 25 2015

			-13.74276825021405443522056419051855107309537215770498560....

G. C. Greubel, Table of n, a(n) for n = 2..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-18], 100]]
```

zeta'(-18) = -(97692469875*zeta(19))/(8*Pi^18) = - log(A(18)).

Equals -(43867/3192)*(zeta(19)/zeta(18)).

Offset corrected by Rick L. Shepherd, May 30 2016

A266274 Decimal expansion of zeta'(-19) (the derivative of Riemann's zeta function at -19) (negated).

Original entry on oeis.org

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

Offset: 2

G. C. Greubel, Dec 26 2015

			-29.965529831392351939431865297274201791908226109115565915881....

G. C. Greubel, Table of n, a(n) for n = 2..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-19], 100]]
```

zeta'(-n) = (BernoulliB(n+1)*HarmonicNumber(n))/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant.

zeta'(-19) = -48069674759189/512143632000 - log(A(19)).

Offset corrected by Rick L. Shepherd, May 30 2016

A266275 Decimal expansion of zeta'(-20) (the derivative of Riemann's zeta function at -20).

Original entry on oeis.org

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

Offset: 3

G. C. Greubel, Dec 26 2015

			132.28099750421251452709821158578551868064800999955031458847450192414...

G. C. Greubel, Table of n, a(n) for n = 3..1500

Cf. A075700 (zeta'(0)), A084448 (zeta'(-1)), A240966 (zeta'(-2)), A259068 (zeta'(-3)), A259069 (zeta'(-4)), A259070 (zeta'(-5)), A259071 (zeta'(-6)), A259072 (zeta'(-7)), A259073 (zeta'(-8)), A266260 (zeta'(-9)), A266261 (zeta'(-10)), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)).

Mathematica
```
RealDigits[N[Zeta'[-20], 100]]
```

zeta'(-20) = (9280784638125*zeta(21))/(8*Pi^20) = - log(A(20)).

Equals (174611/1320)*(zeta(21)/zeta(20)).

Offset corrected by Rick L. Shepherd, May 30 2016

A261506 Decimal expansion of -zeta'(4).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 22 2015

			0.06891126589612537984882936558744082715001637487137...

J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV

Cf. A075700 (0), A073002 (2), A244115 (3).

Cf. A084448 (-1), A240966 (-2), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8).

Flatten[{0, RealDigits[-Zeta'[4], 10, 105][[1]]}]

Sum_{n>=1} log(n) / n^4.

A271173 Decimal expansion of the logarithm of the generalized Glaisher-Kinkelin constant A(7) (negated).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Apr 01 2016

The logarithm of the seventh Bendersky constant.

			-0.010074928748412187918961338073921...

G. C. Greubel, Table of n, a(n) for n = 0..2000

Cf. A259072, A266554.

Join[{0}, RealDigits[(BernoulliB[8]/8)*(EulerGamma + Log[2*Pi] - Zeta'[8]/Zeta[8]), 10, 100] // First]

log(A(7)) = (1/8)*HarmonicNumber(7)*Bernoulli(8) - RiemannZeta'(-7).

log(A(7)) = (Bernoulli(8)/8)*(EulerGamma + log(2*Pi) - Zeta'(8)/Zeta(8)).

A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Apr 23 2016

			zeta'(-1/2) = -0.36085433959994760734742080636395106588485278791863221...

Stanislav Sykora, Table of n, a(n) for n = 0..1000

Values of |zeta'(x)| for various x: A073002 (+2), A075700 (0), A084448 (-1), A114875 (+1/2), A240966 (-2), A244115(+3), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8), A261506 (+4), A266260 (-9), A266261 (-10), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)), A271521 (i).

RealDigits[N[-Zeta'[-1/2], 106]] [[1]] (* Robert Price, Apr 28 2016 *)

PARI
```
-zeta'(-1/2)
```

cp's OEIS Frontend

A266273 Decimal expansion of zeta'(-18) (the derivative of Riemann's zeta function at -18) (negated).

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A266274 Decimal expansion of zeta'(-19) (the derivative of Riemann's zeta function at -19) (negated).

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A266275 Decimal expansion of zeta'(-20) (the derivative of Riemann's zeta function at -20).

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A261506 Decimal expansion of -zeta'(4).

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A271173 Decimal expansion of the logarithm of the generalized Glaisher-Kinkelin constant A(7) (negated).

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Mathematica

Formula

A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

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PARI