  We introduce an "$L$-function" $\mathcal{L}$ built up from the integral
representation of the Barnes' multiple zeta function $\zeta$. Unlike the
latter, $\mathcal{L}$ is defined on a domain equipped with a non-trivial action
of a group $G$. Although these two functions differ from each other, we can use
$\mathcal{L}$ to study $\zeta$. In fact, the transformation formula for
$\mathcal{L}$ under $G$-transformations provides us with a new perspective on
the special values of both $\zeta$ and its $s$-derivative. In particular, we
obtain Kronecker limit formulas for $\zeta$ when restricted to points fixed by
elements of $G$. As an illustration of this principle, we evaluate certain
generalized Lambert series at roots of unity, establishing pertinent
algebraicity results. Also, we express the Barnes' multiple gamma function at
roots of unity as a certain infinite product. It should be mentioned that this
work also considers twisted versions of $\zeta$.

Number Theory