A240966 -id:A240966 - 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.

A324993 Decimal expansion of zeta'(-1, 1/3).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 23 2019

			0.093726201760779427484200899133192867368837286938738021525448092545434...

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 23

Cf. A084448, A240966, A306651, A324994.

evalf(Zeta(1,-1,1/3), 120);
evalf(-Pi/(18*sqrt(3)) - log(3)/72 + Psi(1, 1/3) / (12*sqrt(3)*Pi) - Zeta(1, -1)/3, 120);

RealDigits[Derivative[1, 0][Zeta][-1, 1/3], 10, 120][[1]]
N[With[{k=1}, -Sqrt[3] * (9^k - 1) * BernoulliB[2*k] * Pi / (9^k * 8*k) - 3*BernoulliB[2*k] * Log[3] / 9^k / 4 / k - (-1)^k * PolyGamma[2*k-1, 1/3] / 2 / Sqrt[3] / (6*Pi)^(2*k-1) - (9^k-3)*Zeta'[-2*k+1]/2/9^k], 120]

zetahurwitz'(-1, 1/3) \\ Michel Marcus, Mar 24 2019

Equals -Pi/(18*sqrt(3)) - log(3)/72 + PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) - Zeta'(-1)/3.

A324993 + A324994 = -log(3)/36 - 2*Zeta'(-1)/3.

A324995 Decimal expansion of zeta'(-1, 1/4).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 23 2019

			0.093567868970261061186336071647446310061521086038359540523565680572606...

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 24.

Cf. A084448, A240966, A324996.

evalf(Zeta(1,-1,1/4), 120);
evalf(-Pi/32 + Psi(1, 1/4)/(32*Pi) - Zeta(1,-1)/8, 120);

RealDigits[Derivative[1, 0][Zeta][-1, 1/4], 10, 120][[1]]
N[With[{k=1}, -(4^k-1) * BernoulliB[2*k] * Pi / 4^(k+1)/k + (4^(k-1)-1)*BernoulliB[2*k] * Log[2]/k/2^(4*k-1) - (-1)^k*PolyGamma[2*k-1,1/4] / 4 / (8*Pi)^(2*k-1) - (4^k - 2)*Zeta'[1-2*k]/2^(4*k)], 120]

zetahurwitz'(-1, 1/4) \\ Michel Marcus, Mar 24 2019

Equals -Pi/32 + PolyGamma(1, 1/4)/(32*Pi) - Zeta'(-1)/8.
A324995 + A324996 = -Zeta'(-1)/4.
Equals A006752/(4*Pi) + log(A074962)/8 - 1/96. - Artur Jasinski, Feb 23 2023

A324996 Decimal expansion of zeta'(-1, 3/4) (negated).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 23 2019

			-0.05221258304514832888285615658675114937051199095455909460693510398326...

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 24.

Cf. A084448, A240966, A324995.

evalf(Zeta(1,-1,3/4), 120);
evalf(Pi/32 - Psi(1, 1/4)/(32*Pi) - Zeta(1,-1)/8, 120);

RealDigits[Derivative[1, 0][Zeta][-1, 3/4], 10, 120][[1]]
N[With[{k=1}, (4^k-1) * BernoulliB[2*k] * Pi/4^(k+1)/k + (4^(k-1) - 1) * BernoulliB[2*k] * Log[2]/k/2^(4*k-1) + (-1)^k*PolyGamma[2*k-1,1/4] / 4 / (8*Pi)^(2*k-1) - (4^k-2)*Zeta'[1-2*k]/2^(4*k)], 120]

zetahurwitz'(-1, 3/4) \\ Michel Marcus, Mar 24 2019

Equals Pi/32 - PolyGamma(1, 1/4)/(32*Pi) - Zeta'(-1)/8.

A324995 + A324996 = -Zeta'(-1)/4.

A324994 Decimal expansion of zeta'(-1, 2/3) (negated).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 23 2019

			-0.01396244731237074388034460444140926398857665998812431718484139757490...

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 23

Cf. A084448, A240966, A324993.

evalf(Zeta(1,-1,2/3), 120);
evalf(Pi/(18*sqrt(3)) - log(3)/72 - Psi(1, 1/3) / (12*sqrt(3)*Pi) - Zeta(1,-1)/3, 120);

RealDigits[Derivative[1, 0][Zeta][-1, 2/3], 10, 120][[1]]
N[With[{k=1}, Sqrt[3] * (9^k - 1) * BernoulliB[2*k] * Pi / (9^k * 8*k) - 3*BernoulliB[2*k] * Log[3] / 9^k / 4 / k + (-1)^k * PolyGamma[2*k-1,1/3] / 2 / Sqrt[3] / (6*Pi)^(2*k-1) - (9^k-3)*Zeta'[-2*k+1]/2/9^k], 120]

zetahurwitz'(-1, 2/3) \\ Michel Marcus, Mar 24 2019

Equals Pi/(18*sqrt(3)) - log(3)/72 - PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) - Zeta'(-1)/3.

A324993 + A324994 = -log(3)/36 - 2*Zeta'(-1)/3.

A324997 Decimal expansion of zeta'(-1, 1/6).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 23 2019

			0.070452592367204142475462166806035927785155027545830206477019332868362...

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 25.

Cf. A084448, A240966, A324998.

evalf(Zeta(1,-1,1/6), 120);
evalf(-Pi/(12*sqrt(3)) + log(2)/72 + log(3)/144 + Psi(1, 1/3)/(8*sqrt(3)*Pi) + Zeta(1,-1)/6, 120);

RealDigits[Derivative[1, 0][Zeta][-1, 1/6], 10, 120][[1]]
N[With[{k=1}, -(9^k - 1) * (2^(2*k-1) + 1) * BernoulliB[2*k] * Pi/(8*Sqrt[3]*k*6^(2*k - 1)) + BernoulliB[2*k] * (3^(2*k-1) - 1)*Log[2]/(4*k*6^(2*k - 1)) + BernoulliB[2*k]*(2^(2*k-1) - 1) * Log[3]/(4*k*6^(2*k-1)) - (-1)^k*(2^(2*k-1) + 1) * PolyGamma[2*k-1,1/3] / (2*Sqrt[3]*(12*Pi)^(2*k - 1))+(2^(2*k - 1) - 1)*(3^(2*k - 1) - 1)*Zeta'[1-2*k]/2/6^(2*k-1)], 120]

zetahurwitz'(-1, 1/6) \\ Michel Marcus, Mar 24 2019

Equals -Pi/(12*sqrt(3)) + log(2)/72 + log(3)/144 + PolyGamma(1, 1/3)/(8*sqrt(3)*Pi) + Zeta'(-1)/6.

A324997 + A324998 = log(2)/36 + log(3)/72 + Zeta'(-1)/3.

A324998 Decimal expansion of zeta'(-1, 5/6) (negated).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 23 2019

			-0.09108038124252111457135608855586726925096589284446330158841490231214...

J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 25.

Cf. A084448, A240966, A324997.

evalf(Zeta(1,-1,5/6), 120);
evalf(Pi/(12*sqrt(3)) + log(2)/72 + log(3)/144 - Psi(1, 1/3)/(8*sqrt(3)*Pi) + Zeta(1,-1)/6, 120);

RealDigits[Derivative[1, 0][Zeta][-1, 5/6], 10, 120][[1]]
N[With[{k=1}, ((9^k-1)*(2^(2*k-1) + 1)*BernoulliB[2*k]*Pi/(8*Sqrt[3]*k*6^(2*k-1)) + BernoulliB[2*k]*(3^(2*k-1) - 1) * Log[2]/(4*k*6^(2*k-1)) + BernoulliB[2*k]*(2^(2*k-1)-1) * Log[3]/(4*k*6^(2*k-1)) + (-1)^k*(2^(2*k-1) + 1) * PolyGamma[2*k-1,1/3] / (2*Sqrt[3]*(12*Pi)^(2*k-1)) + (2^(2*k - 1) - 1)*(3^(2*k - 1) - 1)*Zeta'[1-2*k]/2/6^(2*k-1))], 120]

zetahurwitz'(-1, 5/6) \\ Michel Marcus, Mar 24 2019

Equals Pi/(12*sqrt(3)) + log(2)/72 + log(3)/144 - PolyGamma(1, 1/3)/(8*sqrt(3)*Pi) + Zeta'(-1)/6.

A324997 + A324998 = log(2)/36 + log(3)/72 + Zeta'(-1)/3.

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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A324993 Decimal expansion of zeta'(-1, 1/3).

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A324995 Decimal expansion of zeta'(-1, 1/4).

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A324996 Decimal expansion of zeta'(-1, 3/4) (negated).

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