# Which L-functions are not "Langlands-Shahidi L-functions"?

The Langlands-Shahidi method, among other things, obtains certain L-functions from the constant term of Eisenstein series attached to so-called $(G,M)$ pairs, where $G$ is a reductive group, $M$ a Levi subgroup, $N$ the unipotent radical. Particularly, these L-functions are given by the adjoint representation of the dual $^LM$ on the Lie algebra of $^LN$. Among the L-functions that arise are certain standard, symmetric power, exterior power, and Rankin-Selberg L-functions.

It is known that the list of L-functions obtained by this method does not give all the L-functions of interest, for example the spin L-function of quasi-split D4, studied by Hundley and Gan.

The question is the following: which L-functions do not arise from the Langlands-Shahidi method? Are they able to be characterized?

• Where did you find the list? If the list is complete, then anything else is not from Langlands-Shahidi method. Jul 15, 2014 at 22:02
• My list comes from Kim's notes in the Fields lecture notes on Automorphic L-Functions. The list is long but I am not certain that it is complete. Jul 28, 2014 at 0:57
• Can you give a link or reference of Kim's notes? Jul 29, 2014 at 2:18
• bit.ly/1As2UzS pp. 143, 144 Jul 29, 2014 at 2:28
• If you want examples of l functions which were studied by integral representation method but are not covered by Langlands shahidi method, you probably can google David Ginzburg Jul 6, 2016 at 8:51