Referring to Hardy's approximate functional equation for Riemann's Zeta

would anybody know of similar results for the Dirichlet function ?

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

the ?(s) is for a unknown-to-me function, and I am not even sure whether such an approximate functional equation might actually exist ...