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Summary of the Langlands-Shahidi Method (mainly superseded by the above survey)	Summary of the Rankin-Selberg Method (from Daniel's Bump's website)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:

Shahidi's paper Intertwining Operators, L-Functions, and Representation Theory, Notes of the Eleventh KAIST Mathematics Workshop 1996, Taejon, Korea; Edited by Ja Kyung Koo. In dvi or pdf.
Robert Langlands Digital Archive
Rutgers University Mathematics Department
Institute for Advanced Study Mathematics Department
Daniel Bump's Website
William Stein's modular forms tables
The Selberg class of L-functions
L-functions and Random Matrix Theory
Paul Garrett's Vignettes
Tim Dokchitser's packge for computing special values of L-functions
Michael Rubinstein's computational data
Course notes on adeles from Math 574, Spring 2002.

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.