Specifically I am interested in the quotients $$\frac{\zeta'(\rho)}{\zeta'(1\rho)}=2(2\pi)^{\rho1}\Gamma(1\rho)\sin(\pi\rho/2).$$ Obviously they are in $\mathbb{T}$ for all known nontrivial zeros. But how often are these number $\pm 1$? I would find some tables of the derivative at the known zeros rather usefull, or even perhaps tables of the quotients above? I would be very grateful if somebody can provide me with a good reference. Thanks!

To elaborate a little more, here's some Mathematica code:
Here's the output: 


Here is an answer in a few parts.


