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][1] ; the linked Wikipedia page also has the equation spelled out. 


  [1]: http://en.wikipedia.org/wiki/Dirichlet_eta_function