$L$-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.