On Hilbert-Polya conjecture: Hermitian operator naturally associated to L-functions

Using as starting point a classical integral representation of a L-function we define a familly of two variables extended functions which are eigenfunctions of a Hermitian operator (having imaginary part of zeros as eigenvalues). This Hermitian operator can take also other forms, more symetric. In the case of particular L-functions, like Zeta function or Dirichlet L-functions, the eigenfunctions defined for this operator have symmetry properties. Moreover, for s zero fo Zeta function (or Dirichlet L-function), the associated eigenfunction has a specific property (a part of eigenfunction is cancelled). Finding such an eigenfunction, square integrable due to this "cancellation effect", would lead to Riemann Hypothesis using Hilbert-Polya idea.