Dirichlet eta approximate functional equation

[Simpel]

Concerning Hardy-Littlewood approximate functional equation for the $$\zeta$$ function
$$\zeta(s) = \sum_{n\leq x}\frac{1}{n^s} \ + \ \chi(s) \ \sum_{n\leq y}\frac{1}{n^{1-s}} \ + \ O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1})$$
does somebody know of any similar result for the Dirichlet $$\eta$$ function ? where $$\eta (s)$$ is defined as
$$\eta(s) = \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s} = 1-\frac{1}{2^s}+\frac{1}{3^s}-\frac{1}{4^s}+-\ldots$$

 

[Gib Z]

Is something like this what you were looking for?
$$\eta(s) = (1- 2^{1-s}) \left( \sum_{n\leq x}\frac{1}{n^s} \ + \ \chi(s) \ \sum_{n\leq y}\frac{1}{n^{1-s}} \ + \ O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1}) \right)$$

 

[Simpel]

that would be too easy,
but that was my fault, as I should have better described what I meant by "similar".
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 one. So, I am looking for something like this (the ? is for a unknown-to-me function, and I am not even sure that such an approximate functional equation might exist ...) :
$$\eta(s) = \sum_{n\leq x}\frac{(-1)^{n-1}}{n^s} \ + \ ?(s) \ \sum_{n\leq y}\frac{(-1)^{n-1}}{n^{1-s}} \ + \ O( ...)$$
I would also be happy to try out directly on the Dirichlet Eta (if it makes any sense at all) the method followed by Hardy to get the approximate functional equation for the Zeta function, but I have googled around without finding any detailed description of such method, would anybody know a useful reference ?