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International audienceWe consider the semi-classical (or Gutzwiller-Voros) zeta functions for $C^\infty $ contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to the flow, we prove that, for arbitrarily small $\tau >0$, its zeros are contained in the union of the $\tau $-neighborhood of the imaginary axis, $|\mathfrak {R}(s)|<\tau $, and the half-plane $\mathfrak {R}(s)<-\chi _0+\tau $, up to finitely many exceptions, where $\chi _0>0$ is the hyperbolicity exponent of the flow. Further we show that the density of the zeros along the imaginary axis satisfy an analogue of the Weyl law

The semiclassical zeta function for geodesic flows on negatively curved manifolds