Double integral involving zeta function: $\int_0^\infty \frac{1-12y^2}{(1+4y^2)^3}\int_{1/2}^{\infty}\log|\zeta(x+iy)|~dx ~dy.$

Question

I'm having trouble evaluating the following double integral:

$$\int_0^\infty \frac{1-12y^2}{(1+4y^2)^3}\int_{1/2}^{\infty}\log|\zeta(x+iy)|~dx ~dy.$$

Do please remark that $\zeta$ is the zeta function.

I don't even really know where to start. Please offer any hints.

Answer 1

I suspect that this is a troll (especially considering the comment), but regardless;

In 1995, Volchkov proved that the integral you're interested in is equal to $\frac{\pi(3-\gamma)}{32}$ if and only if the Riemann Hypothesis is true.