Another application of the computation of large numbers of Riemann zeros (beyond verification of the Riemann Hypothesis) is towards bounding the deBruijn-Newman constant $\Lambda$:

deBruijn introduced a deformation parameter $t$ in the Riemann $\Xi$ function so that $\Xi_0(x)=\Xi(x)$ and the Riemann zeros $x(t)$ flow according to the "backward heat equation." Together their work shows the existence of a constant $\Lambda$ such that, for $\Lambda\le t$ the function $\Xi_t(x)$ has only real zeros, while for $t<\Lambda$ there exist complex zeros. The Riemann Hypothesis is the conjecture that $\Lambda\le 0$. Newman made the complementary conjecture that $\Lambda\ge 0$, writing "This new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so." Csordas, Smith, and Varga were able to analyze the ODEs governing the motion of the zeros, and use the fact that an very close pair of zeros, so-called Lehmer pairs, would give a lower bound on $\Lambda.$

The current best bound via this approach, due to Saouter, Gourdon, and Demichel, is that
$$
-1.14\times 10^{-11}<\Lambda
$$
based on a Lehmer pair at height about $7.95\times 10^{12}$