Since one has $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = (1 -2^{1-s}) \zeta (s)$$ you get a Functional equation directly from the one for $\zeta$.

Note: This is function is also called Dedekind eta function.

Since one has $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = (1 -2^{1-s}) \zeta (s)$$ you get a Functional equation directly from the one for $\zeta$.

Note: This is function is also called Dirichlet eta function ; the linked Wikipedia page also has the equation spelled out.