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We know, assuming eta function, that
$$\eta(x) =\sum_{n=1}^{inf}(-1)^{1-n}\frac{1}{n^{x}}$$
and
$$\eta (x)=(1-2^{1-x})\zeta (x)$$
so we have zeros of this function at x=1 for
$$\frac{2\pi k}{\ln\left(2\right)}$$
so
$$\Re \eta(x+iy) =\sum_{n=1}^{inf}(-1)^{1-n}\frac{cos(yln(n))}{n^{x}}$$
and
$$\Im ...
