Welcome to the LMFDB, the database of L-functions, modular forms, and related objects. These pages are intended to be a modern handbook including tables, formulas, links, references, etc. to very concrete objects, in particular specific L-functions and their sources. L-functions are ubiquitous in number theory and have applications to mathematical physics and cryptography. By an L-function, we generally mean a Dirichlet series with a functional equation and an Euler product, the simplest example being the Riemann zeta function. Two of the seven Clay Mathematics Million Dollar Millennium Problems deal with properties of these functions, namely the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture. L-functions arise from and encode information about a number of mathematical objects. It is necessary to exhibit these objects along with the L-functions themselves, since typically we need these objects to compute L-functions. In these pages you will see examples of L-functions coming from modular forms, elliptic curves, number fields, and Dirichlet characters, as well as more generally from automorphic forms, algebraic varieties, and Artin representations. In addition, the database contains details about these objects themselves. See the Map of LMFDB for descriptions of connections between these objects. For additional information, there is a useful collection of freely available online sources at http://www.numbertheory.org/ntw/lecture_notes.html. The subject of L-functions is very rich, with many interrelationships. Our goal is to describe the data in ways that faithfully exhibit these interconnections, and to offer access to the data as a means of prompting further exploration and discovery. We believe that the creation of this website will lead to the development and understanding of new mathematics.
Keywords for this software
References in zbMATH (referenced in 10 articles )
Showing results 1 to 10 of 10.
- Cooper, I.A.; Morris, Patrick W.; Snaith, N.C.: Beyond the excised ensemble: modelling elliptic curve $L$-functions with random matrices (2016)
- Dehaye, Paul-Olivier; Iancu, Mihnea; Kohlhase, Michael; Konovalov, Alexander; Lelièvre, Samuel; Müller, Dennis; Pfeiffer, Markus; Rabe, Florian; Thiéry, Nicolas M.; Wiesing, Tom: Interoperability in the OpenDreamKit project: the math-in-the-middle approach (2016)
- Deines, Alyson; Fuselier, Jenny G.; Long, Ling; Swisher, Holly; Tu, Fang-Ting: Generalized Legendre curves and quaternionic multiplication (2016)
- Farmer, David W.; Koutsoliotas, Sally: The second Dirichlet coefficient starts out negative (2016)
- Guitart, Xavier; Masdeu, Marc; Şengün, Mehmet Haluk: Uniformization of modular elliptic curves via $p$-adic periods (2016)
- Hulse, Thomas A.; Kuan, Chan Ieong; Kıral, Eren Mehmet; Lim, Li-Mei: Counting square discriminants (2016)
- Mertens, Michael H.: Eichler-Selberg type identities for mixed mock modular forms (2016)
- Patrikis, Stefan: Deformations of Galois representations and exceptional monodromy (2016)
- Linowitz, Benjamin; Voight, John: Small isospectral and nonisometric orbifolds of dimension 2 and 3 (2015)
- He, Junhua: On a conjecture concerning integral real roots of certain cubic polynomials (2014)