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In this paper,  we  give a reformulation of the Riemann hypothesis for automorphic  L -functions  in terms of orthogonal polynomials which extends the Cardon and Roberts criterion for the classical case of the Riemann zeta function. Actually, for any principal  L -function  L ( s ,  π ) attached to irreducible cuspidal unitary automorphic representation   π  of  GL ( N ),  we show that the generalized Riemann hypothesis with simple zeros is equivalent to the existence of a certain family of orthogonal polynomials { P   n  ( z )} such that

The Cardon and Robert Criterion for the Riemann hypothesis