We have, with $s= \sigma + it $, $$\zeta(s,a) = \Gamma(1-s)I(s,a) $$ if $\sigma > 1$. edit: Here $$ I(s,a):= \frac{1}{2\pi i } \int_C \frac{z^{s-1}e^{az}}{1 -e^z} \,dz $$ where $C$ is the Hankel Integral. (pg253, Apostol, Introduction to Analytic Number Theory).
Here $\Gamma(1-s)$ is not defined when $s \in \mathbb{N}$. What does Apostol mean? He also wrote similar thing, on page 259,
For all $s$ we have $$ \zeta(1-s) = 2 (2 \pi )^{-s} \Gamma(s) \cos \frac{\pi s }{2} \zeta(s) $$
But the Gamma function is not defined for $s \in \{ 0, -1, \ldots , \}$ and $\zeta$ is not defined for $s = 1$?
Recall that $$\zeta(-n)=\frac{B_{n+1}}{n+1},$$ where $B_{m}$ are the Bernoulli numbers, and $B_{2m+1}=0$ for all $m>0$. Also, $\cos(\pi s/2)=0$ for $s<0$ odd.
So, for example, combined we have $$\lim_{s\to-1} \Gamma(s)\cos(\pi s/2)\zeta(s)=\frac{\pi}{24},$$ but taken individually, $\Gamma(-1)=+\infty$, $\zeta(-1)=-1/12$, $\cos(-\pi/2)=0$. In the limit the product of the $\Gamma$ and $\cos$ terms actually converge to a number !
Similarly for your first expression.