A new Bartholdi zeta function of a digraph II

We introduce a new type of the Bartholdi zeta function of a digraph DD. Furthermore, we define a new type of the Bartholdi LL-function of DD, and give a determinant expression of it. We show that this LL-function of DD is equal to the LL-function of DD 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 DD by new Bartholdi LL-functions of DD.