  We show that for a generic conformal metric perturbation of a compact
hyperbolic 3-manifold $\Sigma$ with Betti number $b_1$, the order of vanishing
of the Ruelle zeta function at zero equals $4-b_1$, while in the hyperbolic
case it is equal to $4-2b_1$. This is in contrast to the 2-dimensional case
where the order of vanishing is a topological invariant. The proof uses the
microlocal approach to dynamical zeta functions, giving a geometric description
of generalized Pollicott-Ruelle resonant differential forms at 0 in the
hyperbolic case and using first variation for the perturbation. To show that
the first variation is generically nonzero we introduce a new identity relating
pushforwards of products of resonant and coresonant 2-forms on the sphere
bundle $S\Sigma$ with harmonic 1-forms on $\Sigma$.
