  In this paper we define a symmetric zeta function. We show that it can be
analytically continued to a meromorphic function on $\mathbb{C}^3$ with only
simple poles at some special hyperplanes. We also calculate the value of a
multiple residue at one special point. For a divergent multiple series, which
can be viewed as the value of the symmetric zeta function at the point
$(1,1,1)$, we give a very detailed analysis on its growth and we relate it to
the classical Euler constant.
