  We obtain uniform lower bounds, true for all automorphic L-functions L(s)
associated to cuspidal representations of GL(m,A) where A denotes the adeles of
the rationals Q, of the integral on the vertical line (Re(s)=1/2) of the
absolute value squared of L(s)/s; and also of L(s)/(s-s_0) when s_0 is a zero
of the L-function on the critical line. Several variants are also obtained in
small degrees m, for the vertical integrals at different abscissas in the
critical strip. For the estimates required to prove convergence, we are led to
generalise a result of Friedlander-Iwaniec (Can. J. Math. 57,2005). We obtain
new results on the abscissa of convergence of the L-series. Finally, a problem
is posed about the behaviour of the quadratic integral when s_0 tends to
infinity, in particular for the Riemann zeta function.
