Database of Automorphic L-functions


Maintained by Stephen D. Miller

Rutgers University 

miller@math.rutgers.edu


Last Updated: September 26, 2008



Survey paper "Riemann's Zeta Function and Beyond"
by S. Gelbart and S.D. Miller, Bulletin of the AMS, 41, (2004), 59-112.


Summary of the Langlands-Shahidi Method
(mainly superseded by the above survey)

Summary of the Rankin-Selberg Method
(from Daniel's Bump's website)



Links:


Short table of known results about certain L-functions (warning: under construction -- also, for now certain entries are listed as "unknown" despite the presence of many partial results because of a lack of space and time).  One should not read any significance  into the omissions of an L-function here.   The last column indicates the method of Pairings of Automorphc Distributions introduced by Miller and Schmid here.  Lecture notes on this topic from a conference at NYU can be found here and here also.

Group
Representation of L-group
degree of representation
Integral representation exists?
Langlands-Shahidi method applies?
Complete L-function entire?
Partial L-function entire (taken over unramified nonarchimedean cases)?
Partial L-function entire (including archimedean places)?
Ramified calculations done for integral representation/invariant pairing?
GL(n)
standard
n
Yes: Godement-Jacquet, and later Jacquet-Piatetski-Shapiro-Shalika
Yes
Yes
Yes
Yes
?
GL(n)
symmetric square
n(n+1)/2
Yes: Shimura (n=2, holomorphic forms), Gelbart-Jacquet (general n=2), Piatetski-Shapiro-Patterson (n=3), Bump-Ginzburg (general n); see also Kable
Yes
Unknown
Yes
Unknown
Unknown
GL(n)

(n > 3)
exterior square
n(n-1)/2
Yes: Jacquet-Shalika and Bump-Friedberg

Distributional pairing: Miller-Schmid
Yes
Yes, if n=4 or odd; unknown otherwise
Yes
Yes, over Q
For archimedean spherical principal series (Stade), and invariant pairings for  noncuspidal local representations over Q
GL(n)xGL(m)
tensor product
n*m
Jacquet-Shalika-Piatetski-Shapiro
Yes
Yes (Moeglin-
Waldspurger)
Yes
Yes
For integral representations, thought to be impossible at archimedean place except when m=n-1,n, or n+1; for invariant pairings, Yes for noncuspidal local representations over Q
GL(2) symmetric Cube 4
GL(3) symmetric fourth power 5
GL(3) adjoint 8
SO(7) 2nd fundamental representation of Sp(6) 14
GSO(10) spin
GSO(12) spin
SO(n) x GL(k)
GSp(4) spin 4
GSp(5) standard 5
GSp(6) spin
GSp(6) x GL(2) spin x standard
GSp(8) spin
GSp(10) spin
Sp(2n) x GL(k)
E6 standard
E7 standard
E8
F4 26
G2


This material is based upon work supported by the National Science Foundation under Grant No. 0601009.