Special values of spectral zeta functions of graphs and Dirichlet L-functions

In this paper, we establish relations between special values of Dirichlet $L$-functions and that of spectral zeta functions or $L$-functions of cycle graphs. In fact, they determine each other in a natural way. These two kinds of special values were bridged together by a combinatorial derivative formula obtained from studying spectral zeta functions of the first order self-adjoint differential operators on the unit circle.