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?