On Lerch's formula and zeros of the quadrilateral zeta function

Let $0 < a \le 1/2$ and define the quadrilateral zeta function by $2Q(s,a) := \zeta (s,a) + \zeta (s,1-a) + {\rm{Li}}_s (e^{2\pi ia}) + {\rm{Li}}_s(e^{2\pi i(1-a)})$, where $\zeta (s,a)$ is the Hurwitz zeta function and ${\rm{Li}}_s (e^{2\pi ia})$ is the periodic zeta function. In the present paper, we show that there exists a unique real number $a_0 \in (0,1/2)$ such that $Q(\sigma, a_0)$ has a unique double real zero at $\sigma = 1/2$ when $\sigma \in (0,1)$, for any $a \in (a_0,1/2]$, the function $Q(\sigma, a)$ has no zero in the open interval $\sigma \in (0,1)$ and for any $a \in (0,a_0)$, the function $Q(\sigma, a)$ has at least two real zeros in $\sigma \in (0,1)$. Moreover, we prove that $Q(s,a)$ has infinitely many complex zeros in the region of absolute convergence and the critical strip when $a \in {\mathbb{Q}} \cap (0,1/2) \setminus \{1/6, 1/4, 1/3\}$. The Lerch formula, Hadamard product formula, Riemann-von Mangoldt formula for $Q(s,a)$ are also shown.