I think that Zagier's conjectures on polylogarithms were based on considerable amount of numerical evidence (this has led to impressive work by Beilinson-Deligne, Goncharov...).
Serre's conjecture on modularity of mod $p$ $2$-dimensional Galois representations was also made precise thanks to simultaneous theoretical advances and numerical computations (this has paved the way to the proof of Fermat's last theorem...).
To come back to the prime number theorem, I think that Euler already "proved" it long before Gauss conjectured it by differentiating $\sum_{p\leq x}\frac{1}{p}\sim \log\log x$.
