a directory of all known Lfunctions
[This page is under construction! Contributions are strongly encouraged.]
"It's a whole beautiful subject and the Riemann zeta function is just
the first one of these, but it's just the tip of the iceberg. They are
just the most amazing objects, these Lfunctions  the fact that they
exist, and have these incredible properties are tied up with all these
arithmetical things  and it's just a beautiful subject. Discovering
these things is like discovering a gemstone or something. You're
amazed that this thing exists, has these properties and can do this."
B. Conrey, Dr.
Riemann's Zeros (Atlantic, 2002), p.166
J. Baez, J. Baez, This Week's Finds in Mathematical Physics
week 217
includes very helpful discussion of the Riemann Hypothesis, Extended Riemann Hypothesis, Grand
Riemann Hypothesis, Weil Conjectures, Langlands Programme, the functional equations
of zeta and Lfunctions, modularity of theta functions, etc.
week 218 follows this up,
framing certain issues concerning bewildering array of zeta functions and Lfunctions in
terms of category
theory.
D.W. Farmer, "Modeling families of
Lfunctions" (preprint 11/05)
[abstract:] "We discuss the idea of a 'family of Lfunctions' and describe various methods
which have been used to make predictions about Lfunction families. The methods involve a mixture
of random matrix theory and heuristics from number theory. Particular attention is paid to families of
elliptic curve Lfunctions. We describe two random matrix models for elliptic curve families:
the Independent Model and the Interaction Model."
abelian Lfunctions
J.F. Burnol, "An
adelic causality problem related to abelian Lfunctions",
Journal of Number Theory 87 no.2 (2001) 253269.
Lfunctions of algebraic varieties
Artin Lfunctions
M.R. Murty,
"Selberg's conjectures and Artin Lfunctions",
Bulletin of the AMS (New Series) 31 (1994) 114.
"In his thesis (published in 1920) the German mathematician E. Artin had developed the
arithmetic theory of ‘function fields over a finite field’, in particular the field of
functions on a curve over a finite field, and noted the many similarities with the theory
of number fields developed by Dirichlet, Dedekind, Kronecker and Hilbert. The analogies
between function fields and number fields had been known since Dedekind’s time (at least
in characteristic zero), but Artin’s work was perhaps the first to take the base field to
have positive characteristic as opposed to subfields of the complex numbers. This
required an entirely algebraic development of the subject since the transcendental
techniques derived from working over the field of complex numbers are inapplicable in this
new context. Artin also (later) developed a quite general theory of Lfunctions which,
once again by purely algebraic means, defined functions akin to the zeta function for
general number fields and for function fields. Artin may thus be seen to have been
working to ‘geometrize or at least "algebraicize") number theory’ while Weil was trying to
‘arithmetize geometry’, and Weil has remarked on the excitement with which he and his
colleagues in those days awaited new numbers of the journals which regularly contained
Artin’s work. These results showed that a ‘Riemann hypothesis’ could be formulated for
the Lfunctions arising in function fields. In an astonishing piece of work (announced
in 1940 but only fully written up in 194?) Weil was able to use geometric techniques to
prove the ‘Riemann hypothesis’ for the function field of any curve over a finite field.
n the course of this effort he convinced himself that a reformulation of the foundations
of algebraic geometry was imperative.
His approach to the ‘Riemann hypothesis for finite fields’ was to use the theory of
‘correspondences’ on an algebraic variety which had been developed by Severi and the
‘Italian school’ of algebraic geometers. correspondences on a curve give rise to the
transformations of the associated abelian variety (the Jacobian of a curve) into itself.
eil showed that these could be described by matrices with ladic enties l being a prime
number different from the charateristic of the base field) and that simple algebraic
properties of these matrices were sufficient to prove the results. It is only fitting
that, years later, when systematic theories generalizing Weil’s approach were developed,
they were entitled ‘Weil cohomology theories’."
D. Reed, Figures of Thought (Routledge, New York, 1995)
Artin Lfunctions of graph coverings
A. Terras and H. Stark, Artin LFunctions of Graph Coverings, in Contemporary Math., Vol. 290,
Dynamical, Spectral, and Arithmetic Zeta Functions  Edited by Michel L. Lapidus, and Machiel van
Frankenhuysen (AMS, 2001) 181195
ArtinHecke Lfunctions
A. Weil, "Sur les formules explicites de la théorie des nombres",
Izv. Mat. Nauk (ser. Mat.) 36 (1972)
Lfunctions of automorphic cusp forms
cuspidal Lfunctions
Lfunctions of degree 1
Brian Conrey's Lfunctions page
Lfunctions of degree 2
Brian Conrey's Lfunctions page
Dirichlet Lfunctions
Wikipedia
entry on Dirichlet character and Lseries
Z. Rudnick, "Zeta functions in arithmetic and their spectral
statistics"
Lfunctions of elliptic curves
Charles
Daney's notes, from an accessible explanation of Wiles' proof of Fermat's
Last Theorem
global Lfunctions
global n^{th} symmetric power Lfunctions
Lfunctions of holomorphic modular forms
local Lfunctions
Maass Lfunction
T. Meurman, "On the order of the Maass Lfunction on the
critical line", Number Theory Vol. 1, Budapest Colloq. Math.
Soc., Janos Bolyai 51 (1990)
"Not all Lfunctions are directly associated to arithmetic
or geometric objects. The simplest example of Lfunctions not
of arithmetic/geometric nature are those arising from Maass waveforms
for a Riemann surface X uniformized by an arithmetic subgroup...
of PGL(2,R). They are pullbacks f(z),
to the universal covering space of X, of simultaneous eigenfunctions
for the action of the hyperbolic Laplacian and of the Hecke operators
on X." [E. Bombieri]
Lfunction attached to the modular discriminant
Z. Rudnick, "Zeta functions in arithmetic and their spectral
statistics"
Lfunction attached to eigenfunctions of the Laplacian on the modular domain
Z. Rudnick, "Zeta functions in arithmetic and their spectral
statistics"
Lfunctions of modular forms
Charles
Daney's notes, from an accessible explanation of Wiles' proof of Fermat's
motivic Lfunctions
C. Deninger, "Motivic Lfunctions and
regularized determinants",
Proc. Symp. Pure Math. 55 (1) (1994) 707743.
C. Deninger, "Motivic Lfunctions and
regularized determinants II", from F. Catanese (editor) Arithmetic
Geometry, Symp. Math. 37 (Cortand, 1994)
padic Lfunctions
J. Coates and W. Sinnott, "On padic Lfunctions over
real quadratic fields", Inventiones Mathematicae 25
(1974) 252279.
J. Diamond, "On the values of padic Lfunctions at
positive integers", Acta Arith. 35 (1979) 223237.
B. Ferrero and R. Greenberg, "On the behaviour of padic
Lfunctions at s = 0", Inventiones Mathematicae
50 (1978) 91102.
N.M. Katz, "padic Lfunctions via moduli of elliptic
curves", Proc. Symp. in Pure Math. 29 (1975) 479506.
N.M. Katz, "padic Lfunctions for CM fields",
Inventiones Mathematicae 49 (1978) 199297.
N. Koblitz, "A new proof of certain formulas for padic
Lfunctions", Duke Mathematical Journal 46
(1979) 455468.
principal Lfunctions
quadratic Lfunctions
K. Soundararajan, "Nonvanishing of quadratic Lfunctions at
s = 1/2", Annals of Mathematics 77 (1963) 387405.
RankinSelberg Lfunctions
E. Kowalski, P. Michel, J. Vanderkam, "RankinSelberg Lfunctions
in the level aspect" (preprint, 2000)
Rankin triple Lfunctions
T. Watson, "Central value of the Rankin triple Lfunction for
unramified Maass cuspforms" (?)
Lfunctions from Selberg class S
Brian Conrey's Lfunctions page
standard Lfunctions
symmetric power Lfunctions of GL_{2}
Kim and Shahidi, "Symmetric cube Lfunctions for
GL_{2} are entire", Annals of Mathematics 150
(1999) 645662.
triple product Lfunctions
M. Harris and S. Kudla, "The central value of a triple product
Lfunction", Annals of Mathematics 133 (1991)
605672.
twisted/elliptic Lfunctions
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