Referring to Hardy's approximate functional equation for Riemann's Zeta
 \zeta(s) = \sum_{n\leq x}\frac{1}{n^s} \ + \ \chi(s) \ \sum_{n\leq y}\frac{1}{n^{1-s}} \ + \ O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1})
would anybody know of similar results for the Dirichlet function ?
 \eta(s) = \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s} = 1-\frac{1}{2^s}+\frac{1}{3^s}-\frac{1}{4^s}+-\ldots
I am interested in expressing the Dirichlet eta function in terms of its partial sums, as well as of the partials sums of its critical line symmetrical "twin". So, I am looking for something like this
 \eta(s) = \sum_{n\leq x}\frac{(-1)^{n-1}}{n^s} \ + \ ?(s) \ \sum_{n\leq y}\frac{(-1)^{n-1}}{n^{1-s}} \ + \ O( ........)
the ?(s) is for a unknown-to-me function, and I am not even sure whether such an approximate functional equation might actually exist ...