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

Author

Artur Jasinski, Sep 01 2023

Keywords

• nonn
• cons

Comments

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

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

Examples

			0.37980489179139...
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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]]

PARI

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

Formula

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.