Integral Representations of Riemann auxiliary function

We prove that the auxiliary function $(s)$ has the integral representation $$(s)=-\frac{2^s{\pi }^s{e}^{\pi is\mathrm{/}4}}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}{y}^s\frac{1-e^{-\pi y^2+\pi \omega y}}{1-e^{2\pi \omega y}}\frac{dy}{y},\omega =e^{\pi i\mathrm{/}4},\mathrm{\Re }s>0,$$ valid for $\sigma >0$. The function in the integrand $\frac{1-e^{-\pi y^2+\pi \omega y}}{1-e^{2\pi \omega y}}$ is entire. Therefore, no residue is added when we move the path of integration.