The Dirichlet $eta$-function is defined as:
$$eta(s) = sum_{n=1}^infty frac{(-1)^{n+1}}{n^s} qquad Re(s) > 0$$
and has the full analytical continuation:
$$eta(s) = sum_{n=1}^N frac{(-1)^{n+1}}{n^s} + frac{(-1)^N}{2} int_{-infty}^{infty} frac{(N+1/2 +ix)^{-s}}{cosh(pi x)},dx qquad s in mathbb{C} tag{1}$$
valid for all integers $N ge 0$.
Wondered what would happen for negative $N$ and found numerically that:
$$eta(s) = sum_{n=1}^{-N-1} frac{(-1)^{n+1}}{n^s} + (-1)^{s+1},frac{(-1)^N}{2} int_{-infty}^{infty} frac{(N+1/2 +ix)^{-s}}{cosh(pi x)},dxqquad s in mathbb{Z} tag{2}$$
valid for all integers $N < 0$.
Note: assume the sums to be zero when their end values are $< 1$.
Question:
Is there a way to also expand equation (2) to $s in mathbb{C}$? If possible, I believe this would require some smart continuation of the $(-1)^{s+1}$ factor. Experimented with functions like $cosleft(pi(s+1)right)$, but no success yet.
