Finding a solutions for an equation

nt.quantity concept – Continuing an analytic continuation of the Dirichlet $eta$-duty? retort

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nt.quantity concept – Continuing an analytic continuation of the Dirichlet $eta$-duty?

The Dirichlet $eta$-duty is outlined as:
$$eta(s) = sum_{n=1}^infty frac{(-1)^{n+1}}{n^s} qquad Re(s) > 0$$

and has the whole analytical continuation:

$$eta(s) = sum_{n=1}^N frac{(-1)^{n+1}}{n^s} + frac{(-1)^N}{2} int_{-infty}^{infty} frac{(N+1/2 +ix)^{-s}}{cosh(pi x)},dx qquad s in mathbb{C} tag{1}$$

precise for all integers $N ge 0$.

Wondered what would befall for traverse $N$ and create numerically that:

$$eta(s) = sum_{n=1}^{-N-1} frac{(-1)^{n+1}}{n^s} + (-1)^{s+1},frac{(-1)^N}{2} int_{-infty}^{infty} frac{(N+1/2 +ix)^{-s}}{cosh(pi x)},dxqquad s in mathbb{Z} tag{2}$$

precise for all integers $N < 0$.

level to: occupy the sums to breathe zero when their stop values are $< 1$.

Question:

Is there a path to too broaden equation (2) to $s in mathbb{C}$? If workable, I endure this is able to require some mental continuation of the $(-1)^{s+1}$ issue. Experimented with features affection $cosleft(pi(s+1)capable)$, however no success but.

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