Ruelle mentions ( http://www.ihes.fr/~ruelle/PUBLICATIONS/%5B94%5D.pdf ) Lee and Yang' s "circle theorem", which comes from statistical mechanics and shall have not yet explored connections with zeta functions and Weil conjectures. Does one know now more about that? (Thanks to Alexandre Eremenko for the hint to that interesting article in an other MO thread)
I do not know about any connection with Weil's conjectures, and this should be considered an extended comment rather than an answer.
Actually Ruelle was wrong when he said that this result "remains isolated" in mathematics. A new proof which fits very well into "manstream" mathematics was given in 1981 in MR0623156 (83c:82008) Lieb, Elliott H.; Sokal, Alan D. A general Lee-Yang theorem for one-component and multicomponent ferromagnets. Comm. Math. Phys. 80 (1981), no. 2, 153–179.
It uses a result of Takagi (early 20-s century) on the distribution of zeros of polynomials, which does not seem to be related to physics. Subsequent papers of A. Sokal further explore this idea.