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

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?