According to an answer on this MO post, showing that $$\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{1/2}^{\infty}\log|\zeta(\sigma+it)|~d\sigma ~dt=\frac{\pi(3-\gamma)}{32}$$
$($$\gamma$ is the Euler-Mascheroni constant$)$ is equivalent to the Riemann hypothesis.
I have two questions:
$(1)$ Has any serious attempt been made to evaluate this numerically or determine strong bounds?
$(2)$ Would numerically evaluating this integral be a valid heuristic argument in favour of the Riemann hypothesis?
Certainly, no amount of numerical accuracy constitutes a proof. However, if we show the equality holds to, say, a quadrillion digits or something, it will be true for all sakes and purposes; I doubt any mathematician would then seriously deny the validity of the conjecture.