172 J.W. COGDELL, £-FUNCTIONS FOR GLn
generic. This idea has been initially investigated by Friedberg and Goldberg [20].
Another possible extension would be to adapt the method to include Eisenstein
series on loop groups [21, 22]. Many more twisted £-functions should be attainable
from Eisenstein series on these groups since more combinations of groups occur in
the Levi decomposition of parabolic subgroups in this context.
Another possibility would be to try to control the twisted £-functions involved
by the method of integral representations, as was pursued in this set of notes for
GLn. While there are many integral representations in the literature, with the
exception of GLn there are few if any cases where a complete analysis of the £-
functions has been worked out. The version of the Converse Theorem given in
Theorem 5.3 was originally put forth for use with integral representations.
Of course, possibly the most natural way to extend the method would be to
reduce the amount of twisting needed in the Converse Theorem. For this reason,
the pursuit of Conjecture 5.2 becomes very appealing.