About database of zeros. mpmath can compute the zeta zeros to arbitrary precision and sage
has optional package containing a lot of zeros (though IIRC with not much precision).

You are asking about products over zeros with scaled imaginary parts, but I suppose
such sums are much easier, including finding closed form assuming RH
(computing such sums without RH will be interesting to me).

[More Zeta Functions for the Riemann Zeros](http://ipht.cea.fr/Docspht/articles/t03/078/public/publi.pdf)
explains how to compute:

$$Z_1(\sigma,v) = \sum_{k=1}^\infty (\tau_k^2+v)^{-\sigma}  $$


$$Z_2(s,x) = \sum_{k=1}^\infty (x-\rho)^{-s}  $$

where {$\rho$} = { $\frac12 \pm i \tau_k$ } k=1,2,... = {the Riemann zeros}

Assume RH (this means $\tau_k \in \mathbb{R}$) and suppose you want to compute
$\sum_\rho \frac{1}{1/2+ 2 i \tau_k}$. 

Grouping $\rho, 1-\rho$ gives $\sum_\rho \frac{1/4}{\tau_k^2+1/16}$ which is directly computable
by $Z_1(1,1/16)$.

Modulo errors the last sum is:

$$1/12\\,{\frac {-16\\,\zeta  \left( 3/4 \right) +3\\,\Psi \left( 3/8
 \right) \zeta  \left( 3/4 \right) -3\\,\ln  \left( \pi  \right) \zeta 
 \left( 3/4 \right) +6\\,\zeta'  \left(3/4 \right) }{\zeta  \left( 3/
4 \right) }}$$

The same approach works for other scalings.