How is $L(s, \pi \times \widetilde \pi)$ for $\pi$ as above defined? Likewise $L(s, \pi \times \bar \phi)$?

Thanks. I guess I never mastered how to compute the endomorphism of the pinning, since I've worked mostly with split groups. Can you recommend a specific reference?

(3) I should perhaps mention that the integral representation in the case m=5 is not really complete. The unramified computation is reduced to an identity which we are able to check in some special cases but not prove in general.

(1) If I'm not mistaken, the Flicker result only applies to representations with a supercuspidal component. Which means that-- again, if I'm not mistaken-- even the case m=3 is not completely solved. (2) The integral representations mentioned all involve Eisenstein series which have poles, so they do not give holomorphy automatically. A useful idea for showing that poles of the Eisenstein series are not inherited by the L function is given in Ginzburg-Jiang JNT 82 pp. 256--287. But one still needs to worry about Archimedean and ramified places.

That seems plausible, but I'm looking at the presentation attached to a root system in section 6 of the Steinberg notes, and it looks to me like in the $A_1$ case it would just give $SL_2(k)$ not $SL_2(k)/\pm 1.$