  We prove an approximate functional equation for the central value of the
L-series attached to an irreducible cuspidal automorphic representation of
GL(m) over a number field with unitary central character. We investigate the
decay rate of the terms involved using the analytic conductor of Iwaniec and
Sarnak as a guideline. Straightforward extensions of the results exist for
products of central values. We hope that these formulae will help further
understanding of the central values of principal L-functions, such as finding
good bounds on their various power means, or establishing subconvexity or
nonvanishing results in certain families. A crucial role in the proofs is
played by recent progress on the Ramanujan--Selberg conjectures achieved by
Luo, Rudnick and Sarnak. The bounds at the non-Archimedean places enter through
the work of Molteni.
