This paper contains the following integral (14) for $\zeta(s)$ with $s=\sigma+it$:
$$\zeta(s) = \frac{\pi}{2\,(s-1)}\int_{-\infty}^{+\infty} \frac{(1/2+ix)^{1-s}}{\cosh^2(\pi x)} dx \qquad \sigma > 0$$
Contrary to the restriction on $\sigma$, numerical evidence suggests this integral is actually valid for all $s \in \mathbb{C}$ (except $s=1$).
Is this true?
Added:
The question has been answered in the comments. Just to share this yields these two beautiful integral expressions for entire Dirichlet series that are valid for all $s \in \mathbb{C}$:
\begin{align*} \eta(s) &= \int_{0}^{\infty} \frac{\left(\frac12+ix\right)^{-s}+\left(\frac12-ix\right)^{-s}}{e^{\pi x}+e^{-\pi x}} dx \\ \\ (s-1)\zeta(s) &= \int_{0}^{\infty} \frac{\left(\frac12+i\frac{x}{\pi}\right)^{1-s}+\left(\frac12-i\frac{x}{\pi}\right)^{1-s}}{\cosh(2x)-1} dx \end{align*}