  The exponentially improved large-$a$ expansion for the Hurwitz zeta function
$\zeta(s,a)$ is exploited to examine the expansion of the periodic zeta
function $F(a,s)$ in the upper half-plane of the variable $a$. It is shown that
a double Stokes phenomenon takes place in the vicinity of the positive
imaginary $a$-axis as $|a|\ra\infty$. This is a consequence of the fact that
constituent parts of $F(a,s)$ involve two Hurwitz zeta functions resulting in
two parallel Stokes lines at unit distance apart. Numerical calculations
confirm the theoretical predictions.
