  Let $a, w_1, w_2,\cdot\cdot\cdot, w_r >0$ and $s \in \mathbb{C}$. We put $w=
(w_1,\cdot\cdot\cdot,w_r)$. Then the Barnes $r$-ple zeta function is defined by
$\zeta_r(s, w, a) = \sum_{m_1=0}^{\infty} \cdot\cdot\cdot \sum_{m_r=0}^{\infty}
1/(a+m_1w_1+\cdot\cdot\cdot +m_rw_r)^s$ when $\sigma := \Re(s)>r$. In this
paper, we show that the Barnes double zeta function $\zeta_2(\sigma, w, a)$ has
real zeros in the interval $(1,2)$ if and only if $0< a < (w_1+w_2)/2$ and the
number of such zero is precisely one if $0< a< (w_1+w_2)/2$.
