  We prove that the dynamical zeta function $Z(s)$ associated to $z^2 + c$ with
$c < -3.75$ has essential zero-free strips of size $1/2 +$, that is, for every
$\epsilon > 0$, there exist only finitely many zeros in the strip
$\mathrm{Re}(s) > 1/2 + \epsilon$. We also present some numerical plots of
zeros of $Z(s)$ using the method proposed in Jenkinson-Pollicott.
