During his investigation of zeta Riemann defined the $\xi$ function as $\xi(s):= \Gamma(\frac{s}{2})(s-1)\pi^{-s/2}\zeta(s)$ which is an entire function that is invariant under the substitution $s \to 1-s$. Moreover $\xi$ shares its zeros with Riemann zeta function $\zeta$.

Riemann wanted to write $\xi(s)$ in the form $\xi(0)\prod_{\rho}(1-\frac{s}{\rho})$ where the product is taken over all zeros of the zeta function. How does one prove the convergence of this product?