We consider the Euler product formula $$\sum_{n=1}^\infty \frac{1}{n^s}=\prod_p \frac{1}{1-p^{-s}}$$

I have two questions about this equality:

1)Does the rate of convergence of each side depend on $s$?

2)For given $s$ is the rate of convergence of the left hand side equal to the rate of convergence of the right hand side of the equality?