Generalized zeta function cutoff causes discontinuity

[quytelda]

Version Information:
Julia 1.11.4
SpecialFunctions v2.5.0

zeta(s, z) gives incorrect values in the region $\Re (z) \geq 7 + \Re (s - 1) + |\Im (s) |$. For example:

julia> zeta(-13+7im, 0.5) # should be ~350.218774132674 + 259.188383123717im
-3371.4397140865763 + 2764.0891272966132im

The function jumps sharply at the boundary of the affected region and no longer behaves as expected. You can see the effects in these plots (using z=0.5):

If my understanding is correct, the Euler-Maclaurin expansion used should be well-defined for $\Re(s) \gt -17$ or so. However, I think the initial sum is prematurely truncated to zero terms when the cutoff is checked.