A new Bartholdi zeta function of a digraph II

We introduce a new type of the Bartholdi zeta function of a digraph D . Furthermore, we define a new type of the Bartholdi L -function of D , and give a determinant expression of it. We show that this L -function of D is equal to the L -function of D defined in [H. Mizuno, I. Sato, A new Bartholdi zeta function of a digraph, Linear Algebra Appl. 423 (2007) 498–511]. As a corollary, we obtain a decomposition formula for a new type of the Bartholdi zeta function of a group covering of D by new Bartholdi L -functions of D .