Functional equation and zeros on the critical line of the quadrilateral zeta function

For $0<a\le 1\mathrm{/}2$, we define the quadrilateral zeta function $Q(s,a)$ using the Hurwitz and periodic zeta functions and show that $Q(s,a)$ satisfies Riemann’s functional equation studied by Hamburger, Heck and Knopp. Moreover, we prove that for any $0<a\le 1\mathrm{/}2$, there exist positive constants $A(a)$ and $T_0(a)$ such that the number of zeros of the quadrilateral zeta function $Q(s,a)$ on the line segment from $1\mathrm{/}2$ to $1\mathrm{/}2+iT$ is greater than $A(a)T$ whenever $T\ge T_0(a)$.