A259069 -id:A259069 - cp's OEIS frontend

A243264 Decimal expansion of the generalized Glaisher-Kinkelin constant A(4).

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

Offset: 0

Jean-François Alcover, Jun 02 2014

Also known as the 4th Bendersky constant.

			0.9920479745250402600134369776254433567369...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

G. C. Greubel, Table of n, a(n) for n = 0..2002
Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant.

Cf. A255323, A259069.

Cf. A019727, A074962, A243262, A243263, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.

RealDigits[Exp[-3*Zeta[5]/(4*Pi^4)], 10, 98] // First
RealDigits[Exp[N[(BernoulliB[4]/4)*(Zeta[5]/Zeta[4]), 100]]] // First (* G. C. Greubel, Dec 31 2015 *)

exp(-3*zeta(5)/(4*Pi^4)) \\ Stefano Spezia, Dec 01 2024

A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(4) = exp(-zeta'(-4)) = exp(-3*zeta(5)/(4*Pi^4)).
A(4) = exp((B(4)/4)*(zeta(5)/zeta(4))). - G. C. Greubel, Dec 31 2015

A260660 Decimal expansion of zeta'(-13) (the derivative of Riemann's zeta function at -13).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Nov 13 2015

			0.06374987374457688028603868147333505564882735...

G. C. Greubel, Table of n, a(n) for n = 0..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)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

N[Zeta'[-13]]
Join[{0}, RealDigits[Zeta'[-13], 10, 1500] // First]

PARI
```
zeta'(-13) \\ Altug Alkan, Nov 13 2015
```

zeta'(-n) = (BernoulliB(n+1)*HarmonicNumber(n))/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant, that is the n-th generalized Glaisher constant.
zeta'(-13) = (1145993/4324320) - log(A(13)).
zeta'(-13) = 1145993/4324320 - gamma/12 - log(2*Pi)/12 + 6081075*Zeta'(14) / (8*Pi^14), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 05 2015

A266260 Decimal expansion of zeta'(-9) (the derivative of Riemann's zeta function at -9).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			0.0031301453197885727549257682907854467026693658654815.....

G. C. Greubel, Table of n, a(n) for n = 0..1501

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)), 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)), A266275 (zeta'(-20)).

Join[{0, 0}, RealDigits[Zeta'[-9], 10, 100] // First]
N[Zeta'[-9], 100]

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

zeta'(-9) = 7129/332640 - log(A(9)).

A266270 Decimal expansion of zeta'(-15) (the derivative of Riemann's zeta function at -15).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			-0.400319302807725593843580317520320367201261286266232944284106942....

G. C. Greubel, Table of n, a(n) for n = 0..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)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-15], 100]]
```

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

zeta'(-15) = -4325053069/2940537600 - log(A(15)).

A266272 Decimal expansion of zeta'(-17) (the derivative of Riemann's zeta function at -17).

Original entry on oeis.org

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

Offset: 1

G. C. Greubel, Dec 25 2015

			3.1286453321241578756844526391533305482263390775654797424916577061....

G. C. Greubel, Table of n, a(n) for n = 1..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)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-17], 100]]
```

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

zeta'(-17) = 1848652896341/175991175360 - log(A(17)).

Offset corrected by Rick L. Shepherd, May 21 2016

A266263 Decimal expansion of zeta'(-12) (the derivative of Riemann's zeta function at -12).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			0.06327058334146300059518230123430776751141818475323637667956594567...

G. C. Greubel, Table of n, a(n) for n = 0..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)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Join[{0}, RealDigits[(691/10920)*(Zeta[13]/Zeta[12]), 10, 100] // First]

zeta'(-12) = (-467775*Zeta(13))/(8*Pi^12) = - log(A(12)).

Equals (691/10920)*(zeta(13)/zeta(12)).

A266264 Decimal expansion of zeta'(-14) (the derivative of Riemann's zeta function at -14).

Original entry on oeis.org

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

Offset: 0

G. C. Greubel, Dec 25 2015

			-0.29165772474387352032122400307025066697026303853309083214990....

G. C. Greubel, Table of n, a(n) for n = 0..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)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-14], 100]]
```

zeta'(-14) = - (42567525*zeta(15))/(16*Pi^14) = - log(A(14)).

Equals -(7/24)*(zeta(15)/zeta(14)).

A266271 Decimal expansion of zeta'(-16) (the derivative of Riemann's zeta function at -16).

Original entry on oeis.org

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

Offset: 1

G. C. Greubel, Dec 25 2015

			1.7730256608990963962477873441892944813554198276469991771639173077.....

G. C. Greubel, Table of n, a(n) for n = 1..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)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)).

Mathematica
```
RealDigits[N[Zeta'[-16], 100]]
```

zeta'(-16) = (638512875*zeta(17))/(4*Pi^16) = - log(A(16)).

Equals (3617/2040)*(zeta(17)/zeta(16)).

Offset corrected by Rick L. Shepherd, May 21 2016

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

Original entry on oeis.org

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

cp's OEIS Frontend

A243264 Decimal expansion of the generalized Glaisher-Kinkelin constant A(4).

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A260660 Decimal expansion of zeta'(-13) (the derivative of Riemann's zeta function at -13).

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A266260 Decimal expansion of zeta'(-9) (the derivative of Riemann's zeta function at -9).

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A266270 Decimal expansion of zeta'(-15) (the derivative of Riemann's zeta function at -15).

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A266272 Decimal expansion of zeta'(-17) (the derivative of Riemann's zeta function at -17).

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A266263 Decimal expansion of zeta'(-12) (the derivative of Riemann's zeta function at -12).

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A266264 Decimal expansion of zeta'(-14) (the derivative of Riemann's zeta function at -14).

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A266271 Decimal expansion of zeta'(-16) (the derivative of Riemann's zeta function at -16).

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