OFFSET
0,1
COMMENTS
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.
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.
EXAMPLE
0.37980489179139...
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]]
PROG
(PARI) real(gamma(exp(I*Pi/3))) \\ Michel Marcus, Sep 01 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Sep 01 2023
STATUS
approved