In the wikipedia the Hadamard product for the Riemann's zeta function has two forms. The first one is $$\zeta(s)=\frac{e^{(\log(2\pi)-1-\gamma/2)s}}{2(s-1)\Gamma(1+s/2)}\prod_{\rho}\left(1-\frac{s}{\rho} \right)e^{s/\rho}$$
The second one is simplified to the form $$\zeta(s)=\frac{\pi^{s/2}}{2(s-1)\Gamma(1+s/2)}\prod_{\rho}\left(1-\frac{s}{\rho} \right)$$
So my question is, how to prove the identity $$e^{(\log(2\pi)-1-\gamma/2)s}\prod_{\rho}e^{s/\rho}= \pi^{s/2}$$
without even know the nontrivial roots, $\rho$, of $\zeta(s)$?
Edit 1:
This is equivalent to calculate the value of $$\sum_{\rho}\frac{1}{\rho}$$ But, how?