  We aim to provide explicit formulas for values and derivative values along a
direction of some generalized Hurwitz multizeta functions at nonpositive
integers. We will express these values in terms of Bernoulli polynomials, and
the derivative values in terms of Bernoulli polynomials, of derivative values
at nonpositive integers of the Hurwitz zeta function, and of derivative values
at nonpositive integers of some generalized Barnes zeta functions. Then we will
apply these formulas in order to compute the values and the derivative values
at $s=0$ of the Witten zeta functions $\zeta_{\mathfrak{g}_2}$ and
$\zeta_{\mathfrak{so}(5)}$. As an application, we will find an explicit
asymptotic formula for the number of representations of dimension $n$ of the
exceptional Lie algebra $\mathfrak{g}_2$ using a generalized Meinardus-type
theorem proved by Bridges, Brindle, Bringmann, and Franke.
