In ch. 3 of 'An introduction to the theory of the Riemann Zeta-Function' (S. J. Patterson 1988), it is claimed that $$ s(1-s)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s) = Ae^{Bs}\prod_{\rho \in \mathcal{Z}}\left(\left( 1 - \frac{s}{\rho} \right)e^{s/\rho}\right)$$ for some $A,B \in \mathbb{C}$, where $\mathcal{Z}$ are the zeros of the function at LHS. Also, $A=-1$ is derived by taking the limit $s \to 0$.
Now, Patterson proceeds to calculate $B$, using both the functional equation and the fact that $\rho \in \mathcal{Z}$ implies $\bar\rho,(1-\rho) \in \mathcal{Z}$ to get: $$-e^{Bs}\prod_{\rho \in \mathcal{Z}}\left(\left( 1 - \frac{s}{\rho} \right)e^{s/\rho}\right) = -e^{B(1-s)}\prod_{\rho \in \mathcal{Z}}\left(\left( 1 - \frac{1-s}{1-\rho} \right)e^{(1-s)/(1-\rho)}\right)$$ and that this equality transforms to: $$e^{B(2s-1)}=\prod_{\rho \in \mathcal{Z}}\left(\left(\frac{\rho}{1-\rho} \right)e^{-1/(1-\rho)}e^{-s/(\rho(1-\rho))}\right)$$
The last step is the one I'm having problems with. As term-by-term algebra gives: $$\left(\left(1 - \frac{s}{\rho} \right)e^{s/\rho}\right)^{-1}\left( 1 - \frac{1-s}{1-\rho} \right) e^{(1-s)/(1-\rho)} \\ = \frac{\rho}{\rho - s}\cdot\frac{s - \rho}{1-\rho}\cdot e^{(1-s)/(1-\rho) - s/\rho} \\ = \frac{-\rho}{1-\rho}\cdot e^{1/(1-\rho) - s/(\rho(1-\rho))}$$
I suspect that my naive term-by-term approach is wrong. Where did I make a mistake? How can I show the last (third) identity?