  We examine the exponentially improved asymptotic expansion of the Lerch zeta
function $L(\lambda,a,s)=\sum_{n=1}^\infty \exp (2\pi ni\lambda)/(n+a)^s$ for
large complex values of $a$, with $\lambda$ and $s$ regarded as parameters. It
is shown that an infinite number of subdominant exponential terms switch on
across the Stokes lines $\arg\,a=\pm\pi/2$. In addition, it is found that the
transition across the upper and lower imaginary $a$-axes is associated, in
general, with unequal scales. Numerical calculations are presented to confirm
the theoretical predictions.
