A365319 -id:A365319 - cp's OEIS frontend

A365317 Decimal expansion of real part of Gamma(exp(i*Pi/3)).

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

Offset: 0

Artur Jasinski, Sep 01 2023

For imaginary part of Gamma(exp(i*Pi/3)) see A365318.

For abs(Gamma(exp(i*Pi/3))) see A365319.

			0.37980489179139...

Juan Arias de Reyna and Jan van de Lune, 2013, On the exact location of the non-trivial zeros of Riemann's zeta function, arXiv:1305.3844 [math.NT], 2013, formula (4).
Eric Weisstein's World of Mathematics, Riemann-Siegel Functions.
Wikipedia, Riemann-Siegel theta function.

Cf. A212877, A212878, A212879, A212880, A365318, A365319.

RealDigits[Re[Gamma[Cos[Pi/3] + I Sin[Pi/3]]], 10, 106][[1]]
(* or *)
RealDigits[Sqrt[Pi/Cosh[Pi Sqrt[3]/2]] Cos[2 RiemannSiegelTheta[Sqrt[3]/2] + ArcTan[Tanh[Pi Sqrt[3]/4]] + Sqrt[3] Log[2 Pi]/2], 10, 106][[1]]

real(gamma(exp(I*Pi/3))) \\ Michel Marcus, Sep 01 2023

Equals sqrt(Pi*sech(Pi*sqrt(3)/2))*cos(2*theta(sqrt(3)/2)+(sqrt(3)/2)*log(2*Pi)+arctan(tanh(Pi*sqrt(3)/4))) where theta is Riemann-Siegel theta function.

A365318 Decimal expansion of negative imaginary part of Gamma(exp(i*Pi/3)).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Sep 01 2023

			0.51727909947484...
Gamma(cos(Pi/3) + I*sin(Pi/3)) = 0.37980489179139...-I*0.51727909947484...

Juan Arias de Reyna and Jan van de Lune, 2013, On the exact location of the non-trivial zeros of Riemann's zeta function, arXiv:1305.3844 [math.NT], 2013, formula (4).
Eric Weisstein's World of Mathematics, Riemann-Siegel Functions.
Wikipedia, Riemann-Siegel theta function.

Cf. A212877, A212878, A212879, A212880.

Cf. A365317 (real part), A365319 (abs).

RealDigits[-Im[Gamma[Cos[Pi/3] + I Sin[Pi/3]]], 10, 106][[1]]
(* or *)
RealDigits[Sqrt[Pi/Cosh[Pi Sqrt[3]/2]] Sin[2 RiemannSiegelTheta[Sqrt[3]/2] + ArcTan[Tanh[Pi Sqrt[3]/4]] + Sqrt[3] Log[2 Pi]/2], 10, 106][[1]]

-imag(gamma(exp(I*Pi/3))) \\ Michel Marcus, Sep 01 2023

Equals sqrt(Pi*sech(Pi*sqrt(3)/2))*sin(2*theta(sqrt(3)/2)+(sqrt(3)/2)*log(2*Pi)+arctan(tanh(Pi*sqrt(3)/4))) where theta is Riemann-Siegel theta function.

cp's OEIS Frontend

A365317 Decimal expansion of real part of Gamma(exp(i*Pi/3)).

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