  In 1997 the author found a criterion for the Riemann hypothesis for the
Riemann zeta function, involving the nonnegativity of certain coefficients
associated with the Riemann zeta function. In 1999 Bombieri and Lagarias
obtained an arithmetic formula for these coefficients using the ``explicit
formula'' of prime number theory. In this paper, the author obtains an
arithmetic formula for corresponding coefficients associated with the Euler
product of Hecke polynomials, which is essentially a product of L-functions
attached to weight 2 cusp forms (both newforms and oldforms) over Hecke
congruence subgroups. The nonnegativity of these coefficients gives a criterion
for the Riemann hypothesis for all these L-functions at once.
