In this paper, we show that all real zeros of the bilateral Hurwitz zeta
function Z(s,a):=ζ(s,a)+ζ(s,1−a) with 1/4≤a≤1/2 are on
only the non-positive even integers exactly same as in the case of (2s−1)ζ(s). We also prove that all real zeros of the bilateral periodic zeta
function P(s,a):=Lis​(e2πia)+Lis​(e2πi(1−a))
with 1/4≤a≤1/2 are on only the negative even integers just like
ζ(s). Moreover, we show that all real zeros of the quadrilateral zeta
function Q(s,a):=Z(s,a)+P(s,a) with 1/4≤a≤1/2 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<1/2 is sufficiently
small. The complex zeros of these zeta functions are also discussed when 1/4≤a≤1/2 is rational or transcendental. As a corollary, we show that
Q(s,a) with rational 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 ζ(s).Comment: 12 pages. We changed the title. Some typos are correcte