$\xi: \mathbb{C}-\{0,1\} \longrightarrow \mathbb{C}$, $\xi(s):=\int_1^\infty (t^{(1-s)/2}+t^{s/2})(\sum_{n=1}^\infty e^{-\pi n^2 t})\frac{dt}{t}+\frac{1}{s-1}-\frac{1}{s}$
How to show that the integral is holomorphic? I was already able to show that the integral exists for $\forall s\in\mathbb{C}$.