  We consider the semi-classical (or Gutzwiller-Voros) zeta function for
$C^\infty$ contact Anosov flows. Analyzing the spectrum of transfer operators
associated to the flow, we prove, for any $\tau>0$, that its zeros are
contained in the union of the $\tau$-neighborhood of the imaginary axis,
$|\Re(s)|<\tau$, and the region $\Re(s)<-\chi_0+\tau$, up to finitely many
exceptions, where $\chi_0>0$ is the hyperbolicity exponent of the flow. Further
we show that the zeros in the neighborhood of the imaginary axis satisfy an
analogue of the Weyl law.
