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In our previous work, we established the theory of multi-variable Wittenzeta-functions, which are called the zeta-functions of root systems. We havealready considered the cases of types $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. Inthis paper, we consider the case of $G_2$-type. We define certain analogues ofBernoulli polynomials of $G_2$-type and study the generating functions of themto determine the coefficients of Witten's volume formulas of $G_2$-type. Nextwe consider the meromorphic continuation of the zeta-function of $G_2$-type anddetermine its possible singularities. Finally, by using our previous method, wegive explicit functional relations for them which include Witten's volumeformulas.

On Witten multiple zeta-functions associated with semisimple Lie algebras IV