Riemann xi

The Riemann $\xi$ function is defined by the formula $$\xi(z)=\dfrac{z}{2}(z-1)\pi^{-\frac{z}{2}}\Gamma\left(\dfrac{z}{2}\right)\zeta(z),$$ where $\Gamma$ denotes the gamma function and $\zeta$ denotes the Riemann zeta function.

Properties

Theorem: The values of $\xi$ are known at even integers: $$\xi(2n) = \dfrac{(-1)^{n+1}}{(2n)!}B_{2n}2^{2n-1}\pi^n (2n^2-n)(n-1)!,$$ where $B_n$ is the $n$th Bernoulli number.

Proof: █