Evaluating a Hankel contour of zeta-like function

Question

I was reading a 'paper' about the zeta function (I think the main part is fatally flawed, but there is still some valid work). At one point, the author deduces for $\Re(s)>0$ $$ (s-1)\zeta(s)=\frac{1}{\Gamma(s)}\int_0^{\infty} \phi(t)e^{-t}t^{s-1}\,dt, $$where $\phi(t)= \frac{t}{(1-e^{-t})^2}-\frac{1}{1-e^{-t}}$. I buy this. What I'm not sure about is the following claim: for $\Re(s)\leq 0$, $$ (s-1)\zeta(s) = \frac{\Gamma(1-s)}{2\pi i}\int_{\mathcal{C}} \phi(t)e^{-t}t^{s-1}\,dt, $$Here $\mathcal{C}$ is the Hankel contour from $(-\infty,-\varepsilon)$, around a counterclockwise circle around the origin, and then back. Any advice about this second integral would be greatly appreciated.

https://arxiv.org/abs/1004.4143