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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?

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  • $\begingroup$ Where did you find the list? If the list is complete, then anything else is not from Langlands-Shahidi method. $\endgroup$
    – 7-adic
    Jul 15 '14 at 22:02
  • $\begingroup$ 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. $\endgroup$
    – Tian An
    Jul 28 '14 at 0:57
  • $\begingroup$ Can you give a link or reference of Kim's notes? $\endgroup$
    – 7-adic
    Jul 29 '14 at 2:18
  • $\begingroup$ bit.ly/1As2UzS pp. 143, 144 $\endgroup$
    – Tian An
    Jul 29 '14 at 2:28
  • $\begingroup$ 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 $\endgroup$
    – Q. Zhang
    Jul 6 '16 at 8:51
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The most complete list I could find of L-functions accessible by the Langlands-Shahidi method (for split groups only) is found in the monograph Lectures on Automorphic L-functions.

EDIT: Henry Kim has a paper in which "The purpose of this paper is three-fold; first, to make it explicit all L-functions which appear in the constant term of the Eisenstein series by combining the list in [La] and [Sh3]..", titled On local L-functions and normalized intertwining operators.

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    $\begingroup$ This isn't really an answer, so it should probably have been a comment or an edit to the question. $\endgroup$ May 20 '15 at 9:36
  • $\begingroup$ I happened to be thinking about this question again, and @DavidLoeffler's comment is totally accurate, so I am unaccepting that answer of mine, which indeed should have been a comment. $\endgroup$
    – Tian An
    Jun 12 '20 at 16:56

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