  In this paper we prove that the Generalized Riemann Hypothesis (GRH) for
functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class
is equivalent to a certain integral expression of the real part of the
generalized Li coefficient $\lambda_F(n)$ associated to
$F\in\mathcal{S}^{\sharp\flat}$, for positive integers $n$. Moreover, we deduce
that the GRH is equivalent to a certain expression of $Re(\lambda_F(n))$ in
terms of the sum of the Chebyshev polynomials of the first kind. Then, we
partially evaluate the integral expression and deduce further relations
equivalent to the GRH involving the generalized Euler-Stieltjes constants of
the second kind associated to $F$. The class $\mathcal{S}^{\sharp\flat}$
unconditionally contains all automorphic $L$-functions attached to irreducible
cuspidal unitary representations of $GL_N(\mathbb{Q})$, hence, as a corollary
we also derive relations equivalent to the GRH for automorphic $L$-functions.
