Riemann xi function

Riemann's original lower-case "xi"-function, \xi was renamed with a \Xi (Greek uppercase letter "xi") by Edmund Landau. Landau's \xi (lower-case "xi") is defined as : \xi(s) = \frac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s) for s \in \mathbb{C}. Here \zeta(s) denotes the Riemann zeta function and \Gamma(s) is the gamma function. The functional equation (or reflection formula) for Landau's \xi is : \xi(1-s) = \xi(s) . Riemann's original function, renamed as the upper-case \Xi by Landau, satisfies : \Xi(z) = \xi \left(\tfrac{1}{2} + z i \right) , and obeys the functional equation : \Xi(-z) = \Xi(z) . Both functions are entire and purely real for real arguments. == Values ==

The general form for positive even integers is : \xi(2n) = (-1)^{n+1}\frac{n!}{(2n)!}B_{2n}2^{2n-1}\pi^{n}(2n-1) where B_n denotes the th Bernoulli number. For example: : \xi(2) = {\frac{\pi}{6}} == Series representations ==

The \xi function has the series expansion : \frac{d}{dz} \ln \xi \left(\frac{-z}{1-z}\right) = \sum_{n=0}^\infty \lambda_{n+1} z^n, where : \lambda_n = \frac{1}{(n-1)!} \left. \frac{d^n}{ds^n} \left[s^{n-1} \log \xi(s) \right] \right|_{s=1} = \sum_{\rho} \left[ 1- \left(1-\frac{1}{\rho}\right)^n \right], where the sum extends over \rho, the non-trivial zeros of the zeta function, in order of \vert\Im(\rho)\vert. This expansion plays a particularly important role in Li's criterion, which states that the Riemann hypothesis is equivalent to having \lambda_n > 0 for all positive n. == Hadamard product ==

A simple infinite product expansion is : \xi(s) = \frac12 \prod_\rho \left(1 - \frac{s}{\rho} \right), where \rho ranges over the roots of \xi. To ensure convergence in the expansion, the product should be taken over "matching pairs" of zeroes, i.e., the factors for a pair of zeroes of the form \rho and \bar\rho should be grouped together. == References ==