Numerical calculations on the zeroes of the Riemann zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) or more have been calculated, all simple and all located on the critical line $\Re s=1/2$.
Question 1: what is the order of magnitude of the largest known zero?
Question 2: assuming RH and writing the $k$-th zero as $z_k=\frac12+i t_k$ with a non-decreasing sequence $t_k$ of positive numbers, is there an asymptotic formula for the size of $t_k$ in terms of $k$?
Question 3: if the Riemann hypothesis is wrong, up to a double logarithmic error, say if in fact, no better approximation than
$$
\pi(x)=\text{Li}(x)+O\bigl(x^{1/2}(\ln x)( \ln \ln x)\bigr),
$$
holds true, how far do the numerical computations should go to detect that "doubly logarithmic" quantity of zeroes off the critical line? If it is up to $10^{100}$, there is a chance that in the next 30 years a computer can detect a zero off line. If it is of the order of $10^{10000}$, RH won't be disproved by a computer before the sun becomes a red giant.
Best Answer
Regarding Question 1, for their paper The zeta function on the critical line: numerical evidence for moments and random matrix theory models, Hiary and Odlyzko computed 5 billion zeros near the $10^{23}$rd zero. The last had imaginary part approximately $$ 1.30664344087959822199974045053551×10^{22} $$
See Table 2 of http://www.dtc.umn.edu/~odlyzko/doc/zeta.moments.pdf
This seems to be the current record.
Update: In “Alan Turing and the Riemann Zeta Function” by Hejhal and Odlyzko, in the book Alan Turing - His Work and Impact, Elsevier 2013, they write “It is now known that the RH is true for … some hundreds of zeros near zero number $10^{32}$” (This is $t$ near $9.04808\cdot 10^{30}$.)