Zeros and the functional equation of the quadrilateral zeta function

In this paper, we show that all real zeros of the bilateral Hurwitz zeta function $Z(s,a):=\zeta (s,a)+\zeta (s,1-a)$ with $1\mathrm{/}4\le a\le 1\mathrm{/}2$ are on only the non-positive even integers exactly same as in the case of $(2^s-1) \zeta (s)$. We also prove that all real zeros of the bilateral periodic zeta function $P(s,a):={\mathrm{L}\mathrm{i}}_s(e^{2\pi ia})+{\mathrm{L}\mathrm{i}}_s(e^{2\pi i(1-a)})$ with $1\mathrm{/}4\le a\le 1\mathrm{/}2$ are on only the negative even integers just like $\zeta (s)$. Moreover, we show that all real zeros of the quadrilateral zeta function $Q(s,a):=Z(s,a)+P(s,a)$ with $1\mathrm{/}4\le a\le 1\mathrm{/}2$ are on only the negative even integers. On the other hand, we prove that $Z(s,a)$, $P(s,a)$ and $Q(s,a)$ have at least one real zero in $(0,1)$ when $0<a<1\mathrm{/}2$ is sufficiently small. The complex zeros of these zeta functions are also discussed when $1/4 \le a \le 1/2$isrationalortranscendental\mathrm{.}Asacorollary,weshowthat$Q(s,a)$withrational$1/4 < a < 1/3$or$1/3 < a < 1/2$ does not satisfy the analogue of the Riemann hypothesis even though $Q(s,a)$ satisfies the functional equation that appeared in Hamburger’s or Hecke’s theorem and all real zeros of $Q(s,a)$ are located at only the negative even integers again as in the case of $\zeta (s)$.