In the first chapter, we recall and study the main classical results of the Riemann zeta function. Then in the second chapter, we focus on the connection between this latter and the prime numbers, by showing the Riemann explicit formula, and by showing for completeness the Von Mangoldt explicit formula. Finally in the last chapter, we first see the preliminary notions necessary to introduce the quite recent definition of the spectral zeta function of a graph and we recall in details some of the results already found. We will then use the first and second chapters to find similar structures and analogies between the Riemann zeta function and the spectral zeta function. In particular, we will make new graphs of these functions, find some new formulas and conjectures, and find some new special values and verify those already found.