  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.
