A365318 - OEIS

%I #25 Oct 05 2023 14:28:49

%S 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,

%T 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,

%U 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

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

%H Juan Arias de Reyna and Jan van de Lune, 2013, <a href="https://arxiv.org/abs/1305.3844">On the exact location of the non-trivial zeros of Riemann's zeta function</a>, arXiv:1305.3844 [math.NT], 2013, formula (4).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Riemann-SiegelFunctions.html">Riemann-Siegel Functions</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_theta_function">Riemann-Siegel theta function</a>.

%F 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.

%e 0.51727909947484...

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

%t RealDigits[-Im[Gamma[Cos[Pi/3] + I Sin[Pi/3]]], 10, 106][[1]]

%t (* or *)

%t 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]]

%o (PARI) -imag(gamma(exp(I*Pi/3))) \\ _Michel Marcus_, Sep 01 2023

%Y Cf. A212877, A212878, A212879, A212880.

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

%K nonn,cons

%O 0,1

%A _Artur Jasinski_, Sep 01 2023