The Riemann-Siegel integral formula is the following representation of the xi-function ξ(s) found in Riemann's Nachlass by Bessel-Hagen in 1926. The formula is essentially (2ξ(s))/(s(s - 1)) = F(s) + (F(1 - s^_))^_, where F(s) = Γ(1/2 s) π^(-s/2) integral_(0↘1) (e^(-i n x^2) x^(-s) d x)/(e^(i π x) - e^(-i π x)),

xi-function

Bernhard Riemann | Carl Ludwig Siegel