Hi everyone.

Could you plz tell me where the zeros of $f(s)$ in the strip $\{0 < \Re s < 1 \}$ are ?

Do they all have $\Re s= 1/2$ ?

$$f(s) = 1 - 2^{-s} - 3^{-s} + 4^{-s} - 5^{-s} + ...$$

$$= \sum a(n)/n^s$$

with $a(n) = -1\\ $ if $n = -1 \mod 3\\ $ or $n = -1 \mod 4,\\ $ $a(n) = 1\\ $ otherwise.

$$f(s) = \zeta(s) - 2 \big[ 3^{-s} \zeta(s,-1/3) + 4^{-s} \zeta(s,-1/4) - 12^{-s} \zeta(s,-1/12) \big]$$

Could you plz give me some zeros in the strip $\{0 < \Re z < 1 \}$?

Thanks.

Eta.