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.

References in zbMATH (referenced in 103 articles )

Showing results 1 to 20 of 103.
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  1. Akbary, Amir; Francis, Forrest J.: Euler’s function on products of primes in a fixed arithmetic progression (2020)
  2. Battistoni, Francesco: On small discriminants of number fields of degree 8 and 9 (2020)
  3. Berčič, Katja; Vidali, Janoš: DiscreteZOO: a fingerprint database of discrete objects (2020)
  4. Chung, Hee-Joong; Kim, Dohyeong; Kim, Minhyong; Park, Jeehoon; Yoo, Hwajong: Arithmetic Chern-Simons theory. II (2020)
  5. Daniels, Harris B.; González-Jiménez, Enrique: On the torsion of rational elliptic curves over sextic fields (2020)
  6. Demark, David; Hindes, Wade; Jones, Rafe; Misplon, Moses; Stoll, Michael; Stoneman, Michael: Eventually stable quadratic polynomials over (\mathbbQ) (2020)
  7. Dieulefait, Luis Victor; Soto, Eduardo: Raising the level at your favorite prime (2020)
  8. Dummit, David S.: A note on the equivalence of the parity of class numbers and the signature ranks of units in cyclotomic fields (2020)
  9. Fonseca, Tiago J.: On coefficients of Poincaré series and single-valued periods of modular forms (2020)
  10. Fredrik Johansson: FunGrim: a symbolic library for special functions (2020) arXiv
  11. Garcia-Fritz, Natalia; Pasten, Hector: Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points (2020)
  12. González-Jiménez, Enrique; Najman, Filip: Growth of torsion groups of elliptic curves upon base change (2020)
  13. Jeon, Daeyeol; Schweizer, Andreas: Torsion of rational elliptic curves over different types of cubic fields (2020)
  14. Kim, Chan-Ho; Kim, Myoungil; Sun, Hae-Sang: On the indivisibility of derived Kato’s Euler systems and the main conjecture for modular forms (2020)
  15. Mascot, Nicolas: Hensel-lifting torsion points on Jacobians and Galois representations (2020)
  16. Tengan, Eduardo: Indecomposability and cyclicity in the (p)-primary part of the Brauer group of a (p)-adic curve (2020)
  17. Ahlgren, Scott; Dunn, Alexander: Maass forms and the mock theta function (f(q)) (2019)
  18. Bary-Soroker, Lior; Stix, Jakob: Cubic twin prime polynomials are counted by a modular form (2019)
  19. Bennett, Michael A.; Gherga, Adela; Rechnitzer, Andrew: Computing elliptic curves over (\mathbbQ) (2019)
  20. Berčič, Katja; Kohlhase, Michael; Rabe, Florian: Towards a unified mathematical data infrastructure: database and interface generation (2019)

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