A155968 -id:A155968 - cp's OEIS frontend

A256615 Decimal expansion of log(Gamma(1/24)).

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

Offset: 1

Iaroslav V. Blagouchine, Apr 04 2015

			3.155402877381144722774664455739805690458351588844024...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. decimal expansions of log(Gamma(1/k)): A155968 (k=2), A256165 (k=3), A256166 (k=4), A256167 (k=5), A255888 (k=6), A256609 (k=7), A255306 (k=8), A256610 (k=9), A256612 (k=10), A256611 (k=11), A256066 (k=12), A256614 (k=16), A256616 (k=48).

Maple
```
evalf(log(GAMMA(1/24)),100);
```

RealDigits[Log[Gamma[1/24]],10,100][[1]]

PARI
```
log(gamma(1/24))
```

A256616 Decimal expansion of log(Gamma(1/48)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Apr 04 2015

			3.859529085168528678772669493173125038058701527316499...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. decimal expansions of log(Gamma(1/k)): A155968 (k=2), A256165 (k=3), A256166 (k=4), A256167 (k=5), A255888 (k=6), A256609 (k=7), A255306 (k=8), A256610 (k=9), A256612 (k=10), A256611 (k=11), A256066 (k=12), A256614 (k=16), A256615 (k=24).

Maple
```
evalf(log(GAMMA(1/48)),100);
```

RealDigits[Log[Gamma[1/48]],10,100][[1]]

PARI
```
log(gamma(1/48))
```

A257958 Decimal expansion of the Digamma function at 1/Pi, negated.

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting series representation with rational coefficients for Psi(1/Pi) = -log(Pi) - Pi/2 - 1/2 - 1/8 - 1/72 + 1/64 +7/400 + 7/576 + 643/94080 + 103/30720 + ...

			-3.2902139601732240908430908455400190374021932820070161...

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257955, A257957, A257959, A155968, A256165, A256166, A256167, A255888, A256609, A255306, A256610, A256612, A256611, A256066, A256614, A256615, A256616.

Maple
```
evalf(Psi(1/Pi), 120);
```

RealDigits[PolyGamma[1/Pi], 10, 120][[1]]

PARI
```
default(realprecision, 120); psi(1/Pi)
```

Int_0^infinity x*dx/[(x^2+1)(exp(2x)-1)] = -Pi/2-Psi(1/Pi) = -1.5707...+ 3.2902.. = 1.71941... - R. J. Mathar, Aug 14 2023

A257959 Decimal expansion of the Digamma function at 1/2 + 1/Pi, negated.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting series representation with rational coefficients for Psi(1/2 + 1/Pi) = -log(Pi) + 1/4 + 1/16 - 5/576 - 13/512 - 569/25600 -539/36864 - 98671/12042240 - 16231/3932160 - ...

			-0.9236326759613377346000263347486747139894893215261027...

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257955, A257957, A257958, A155968, A256165, A256166, A256167, A255888, A256609, A255306, A256610, A256612, A256611, A256066, A256614, A256615, A256616.

Maple
```
evalf(Psi(1/2+1/Pi), 120);
```

RealDigits[PolyGamma[1/2+1/Pi], 10, 120][[1]]

default(realprecision, 120); psi(1/2+1/Pi)

A257957 Decimal expansion of log(Gamma(1/Pi)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting series representation with rational coefficients for log(Gamma(1/Pi)) = (1-1/Pi)*log(Pi) - 1/Pi + log(2)/2 + (1 + 1/4 + 1/12 + 1/32 + 1/75 + 1/144 + 13/2880 + 157/46080 + ...)/(2*Pi).

The value log(Gamma(1/Pi)) is also intimately related to integral_{x=0..1} arctan(arctanh(x))/x (A257963).

			1.0336461257655827064855374553316178667100308701595988...

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257955, A257963, A257958, A257959, A155968, A256165, A256166, A256167, A255888, A256609, A255306, A256610, A256612, A256611, A256066, A256614, A256615, A256616.

Maple
```
evalf(log(GAMMA(1/Pi)), 120);
```

RealDigits[Log[Gamma[1/Pi]], 10, 120][[1]]

default(realprecision, 120); log(gamma(1/Pi))

A257963 Decimal expansion of the Integral_{x=0..1} arctan(arctanh(x))/x.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, May 14 2015

"The arctangent of the hyperbolic arctangent is analytic in the whole disk |x| < 1, and therefore, can be expanded into the MacLaurin series", see the first reference.

			= 1.02576051093133045039854866096955279533487185621506939422338684401585192089...

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
Iaroslav V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5 PDF file

Cf. A257957, A257955, A257958, A257959, A155968, A049541, A000796.

evalf(Pi*(log(GAMMA(1/Pi)) - log(GAMMA(1/2 + 1/Pi)) - log(Pi)/2),120); # Vaclav Kotesovec, May 17 2015

nn = 111; RealDigits[ NIntegrate[ ArcTan[ ArcTanh[ x]]/x, {x, 0, 1}, AccuracyGoal -> nn, WorkingPrecision -> nn], 10, nn][[1]] (* or *)
RealDigits[Pi (Log[Gamma[1/Pi]] - Log[Gamma[1/2 + 1/Pi]] - Log[Pi]/2), 10, 111][[1]] (* Robert G. Wilson v, May 14 2015 *)

The integral is equivalent to Pi*(log(Gamma(1/Pi)) - log(Gamma(1/2 + 1/Pi)) - log(Pi)/2), see page 82 of the second reference.

A380908 Decimal expansion of lim_{s->1} (zeta(s) - Pi^(s/2)/((s-1)*Gamma(s/2))) (negated).

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Mar 04 2025

This limit is Mathlib's definition of the 'completed Riemann zeta function' at s = 1. Mathematically zeta(1) is undefined; for the 'completed zeta function' the above limit value is assigned by construction (see Loeffler&Stoll). Such a value is also called a 'junk value' in several proof systems.

			-0.976904291033878966185689752093504708378...

Paolo Xausa, Table of n, a(n) for n = 0..10000
David Loeffler and Michael Stoll, Formalizing zeta and L-functions in Lean, arXiv:2503.00959 [math.NT], March 2025.
Mathlib4, Value of the completed Riemann zeta at s = 1.
Wikipedia, Lean (proof assistant), March 2025.

Cf. A001620, A114864, A155968.

c := -(gamma - log(4*Pi))/2: evalf(c, 110)*10^95: ListTools:-Reverse(convert(floor(%), base, 10));

First[RealDigits[(EulerGamma - Log[4*Pi])/2, 10, 100]] (* Paolo Xausa, Mar 05 2025 *)

(Euler-log(4*Pi))/2 \\ Charles R Greathouse IV, Sep 03 2025

Equals (gamma - log(4*Pi))/2.

Equals gamma + Psi(1/2)/2 - log(Pi^(1/2)) = A001620 - A114864 - A155968.

cp's OEIS Frontend

A256615 Decimal expansion of log(Gamma(1/24)).

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A256616 Decimal expansion of log(Gamma(1/48)).

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A257958 Decimal expansion of the Digamma function at 1/Pi, negated.

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A257959 Decimal expansion of the Digamma function at 1/2 + 1/Pi, negated.

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A257957 Decimal expansion of log(Gamma(1/Pi)).

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Maple

Mathematica

PARI

A257963 Decimal expansion of the Integral_{x=0..1} arctan(arctanh(x))/x.

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Programs

Maple

Mathematica

Formula

A380908 Decimal expansion of lim_{s->1} (zeta(s) - Pi^(s/2)/((s-1)*Gamma(s/2))) (negated).

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