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 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.