  We consider an ``enhanced symmetric space'', which is a prehomogeneous vector
space. This vector space is intimately related to a double flag variety studied
in \cite{NO.2018}. On a distinguished open orbit called ``enhanced positive
cone'', we consider a zeta integral with two complex variables, which is
analytically continued to meromorphic family of tempered distributions. One of
the main results of this paper is to establish a precise formula for the
meromorphic continuation which clarifies the location of poles (and may be
useful to obtain residues). We also compute the Fourier transform of the zeta
distribution and obtain a functional equation with explicit gamma factors.
