Computing the zeta function from its poles and zeros

Question

I'm trying a computational experiment where I am computing $\zeta(s)$ for $s= \frac{1}{2}+ iw$ by separately computing

1. The contribution of all the non-trivial zeros (locations obtained from tables found on the web);

2. The pole at $s=1,$. and

3. The contribution of all the trivial zeros at negative even integers, $$\prod_{n=1}^{a\ very\ large\ number}(1 + \frac{s}{2n})$$ (a very large number limited by computing power).

Unfortunately, this last expression for the contribution of the trivial zeros does not converge for $s=\frac{1}{2} + iw.$

Are there any tricks to circumvent this problem?