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In this paper, we show the following;  (1) The periodic zeta function ${\rm{Li}}_s (e^{2\pi ia})$ with $0<a<1/2$ or$1/2 < a <1$ does not vanish on the real line.  (2) All real zeros of $Y(s,a):=\zeta (s,a) - \zeta (s,1-a)$, $O(s,a) := -i{\rm{Li}}_s (e^{2\pi ia}) + i{\rm{Li}}_s (e^{2\pi i(1-a)})$ and $X(s,a) :=Y(s,a) + O(s,a)$ with $0 < a < 1/2$ are simple and only at the negative oddintegers.  (3) All real zeros of $Z(s,a):=\zeta (s,a) + \zeta (s,1-a)$ are simple andonly at the non-positive even integers if and only if $1/4 \le a \le 1/2$.  (4) All real zeros of $P(s,a):={\rm{Li}}_s (e^{2\pi ia}) + {\rm{Li}}_s(e^{2\pi i(1-a)})$ are simple and only at the negative even integers if andonly if $1/4 \le a \le 1/2$.  Moreover, the asymptotic behavior of real zeros of $Z(s,a)$ and $P(s,a)$ arestudied when $0 < a < 1/4$. In addition, the complex zeros of these zetafunctions are also discussed when $0 <a <1/2$ is rational or transcendental.

On zeros of bilateral Hurwitz and periodic zeta and zeta star functions