**Summary.**  Search for "Weil formules explicites".

Jonathan says in a comment that he is looking for a statement by Weil "that explicitly mentions prime numbers".  This reminds me of his paper *Sur les "formules explicites" de la théorie des nombres premiers.*
Comm. Sém. Math. Univ. Lund (1952). Tome Supplementaire, 252–265.

The review of this paper in [Math Reviews][1] says :

> The most striking result of the paper
> is as follows. The author defines a
> distribution (too complicated to
> define here) whose positivity is
> equivalent to the simultaneous truth
> of the Riemann hypothesis for the
> Artin-Hecke $L$-series and the Artin
> conjecture on their entirety. This
> situation is analogous to the case of
> curves over finite fields for which
> the Riemann hypothesis is a
> consequence of the positivity of the
> trace in the ring of correspondences.

Let me also mention a paper by Burnol in the *Comptes Rendus*, of which the review says

> As is known, the proof of A. Weil of
> the analog for algebraic curves of the
> Riemann hypothesis (R.H.) relies upon
> the equivalence of this hypothesis
> with the positivity of a suitable
> Hermitian form. Weil, again, remarked
> that also the original R.H. for
> $L(s,\chi)$ (the $L$-function
> associated to the Dirichlet character
> $\chi$) holds if and only if $Z(g\ast
> g^\tau)=\sum_{\rho}\widehat{g}(\rho)\overline{\widehat{g}(\overline{1-\rho})}\geq
> 0$ for every smooth compactly
> supported $g$, where $\rho$ runs over
> the critical zeros of $L(s,\chi),\
> \widehat{g}$ is the Mellin transform
> of $g$ and
> $g^\tau(u)=\overline{u^{-1}g(u^{-1})}$.


[1]:http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=weil&s5=formules%2520explicites&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=53152