I'm crossposting.

Being \begin{equation*} \eta(s)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^{s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots \end{equation*} and following Riemann's second method, (Edwards p.15), to obtain the functional equation for $\zeta(s)$ one can think the same way and try the same aproach for $\eta(s)$. Thus from \begin{equation*} \int_{0}^{\infty} \operatorname{exp}\left(-n^{2} \pi x\right) x^{s / 2-1} d x=\pi^{-s / 2} \Gamma\left(\frac{s}{2}\right)\frac{1}{n^{s}} \text { for } s>0 \end{equation*} one can express $\eta(s)$ as \begin{equation*} \pi^{-s / 2} \Gamma \left(\frac{s}{2}\right)\underbrace{\left(1-\frac{1}{2^{s}}+\frac{1}{3^{s}}+\cdots\right)}_{\eta(s)} =\int_{0}^{\infty}\left(e^{-\pi 1^2 x}-e^{-\pi 2^2 x}+e^{-\pi 3^2 x}+\cdots\right)x^{s/2}\text{ }\frac{dx}{x} \end{equation*} How would one proceed from here to craft a functional equation for $\eta(s)$?

I'm interested in refferences and/or answers. Any of them will be very much apreciated.

Thanks.