Perhaps one of the best forms of justification for pure mathematics, in my experience, is the ability to demonstrate the truth of some statements despite the lack of numerical evidence.

A rather striking example that I know of is the story of the average rank of elliptic curves over $\mathbb{Q}$. Goldfeld had conjectured that the average rank ought to be $1/2$ for quadratic twists of a fixed curve, and Katz-Sarnak extended this conjecture to all elliptic curves. However, available numerical evidence in the 1990s when this conjecture was made definitely did not suggest that this should be the case; indeed, curves with low height tend to have larger rank. In 2010 (published in 2015), Bhargava and Shankar made the first substantial step towards the Katz-Sarnak conjecture by showing unconditionally that the average rank is bounded by $3/2$. This was still slightly lower than the numerical average rank at the time. Over two other papers, Bhargava and Shankar eventually obtained an upper bound of $0.877$ for the average rank, but it wasn't until 2016, after computing the rank of millions of elliptic curves, was the numerical average rank shown to be below this theoretical value.

Another older example is Littlewood's disproof of the assertion that $\text{Li}(x) \geq \pi(x)$ for all $x > 1$, which as far as I know is still 'numerically true' in the sense that no one has computed an explicit example of the reverse inequality. Indeed, the upper bound for the smallest counter example to the inequality gave rise to the infamous Skewes' constant.

Are there other theorems which have been proved, but for which have yet to be 'numerically verified'?