  Let $0 < a \le 1$, $s,z \in {\mathbb{C}}$ and $0 < |z|\le 1$. Then the
Hurwitz-Lerch zeta function is defined by $\Phi (s,a,z) := \sum_{n=0}^\infty
z^n(n+a)^{-s}$ when $\sigma :=\Re (s) >1$. In this paper, we show that the
Hurwitz zeta function $\zeta (\sigma,a) := \Phi (\sigma,a,1)$ does not vanish
for all $0 <\sigma <1$ if and only if $a \ge 1/2$. Moreover, we prove that
$\Phi (\sigma,a,z) \ne 0$ for all $0 <\sigma <1$ and $0 < a \le 1$ when $z \ne
1$. Real zeros of Hurwitz-Lerch type of Euler-Zagier double zeta functions are
studied as well.
