$L$-function invariants for 3-manifolds and relations between generalized Bernoulli polynomials

We introduce $L$-functions attached to negative definite plumbed manifolds as the Mellin transforms of homological blocks. We prove that they are entire functions and their values at $s=0$ are equal to the Witten–Reshetikhin–Turaev invariants by using asymptotic techniques developed by the author in the previous papers. We also prove that linear relations between special values at negative integers of some $L$-functions, which are common generalizations of Hurwitz zeta functions, Barnes zeta functions and Epstein zeta functions.