  We consider the dynamical zeta functions of Selberg and Ruelle associated
with the geodesic flow on a compact odd-dimensional hyperbolic manifold. These
dynamical zeta functions are defined for a complex variable $s$ in some
right-half plane of $\mathbb{C}$. In [Spi18], it was proved that they admit a
meromorphic continuation to the whole complex plane. In this paper, we
establish functional equations for them, relating their values at $s$ with
those at $-s$. We prove also a determinant representation of the zeta
functions, using the regularized determinants of certain twisted differential
operators.
