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 158 articles )

Showing results 1 to 20 of 158.
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  1. Amir, Malik; Hong, Letong: On (L)-functions of modular elliptic curves and certain (K3) surfaces (2022)
  2. Asif, Sualeh; Fité, Francesc; Pentland, Dylan: Computing (L)-polynomials of Picard curves from Cartier-Manin matrices (2022)
  3. Barrios, Alexander J.: Minimal models of rational elliptic curves with non-trivial torsion (2022)
  4. Chamizo, Fernando: The additive problem for the number of representations as a sum of two squares (2022)
  5. Cremona, John E.; Freitas, Nuno: Global methods for the symplectic type of congruences between elliptic curves (2022)
  6. Hasanalizade, Elchin; Shen, Quanli; Wong, Peng-Jie: Counting zeros of the Riemann zeta function (2022)
  7. Ohtani, Sachiko: Kummer-faithful fields which are not sub-(p)-adic (2022)
  8. Akbary, Amir; Wong, Peng-Jie: On the moments of torsion points modulo primes and their applications (2021)
  9. Altug, S. Ali; Shankar, Arul; Varma, Ila; Wilson, Kevin H.: The number of (D_4)-fields ordered by conductor (2021)
  10. Balakrishnan, Jennifer S.; Besser, Amnon; Bianchi, Francesca; Müller, J. Steffen: Explicit quadratic Chabauty over number fields (2021)
  11. Banwait, Barinder S.: Examples of abelian surfaces failing the local-global principle for isogenies (2021)
  12. Barquero-Sanchez, Adrian; Mantilla-Soler, Guillermo; Ryan, Nathan C.: Theta series and number fields: theorems and experiments (2021)
  13. Battistoni, Francesco: A conjectural improvement for inequalities related to regulators of number fields (2021)
  14. Benjamin, Nathan; Collier, Scott; Fitzpatrick, A. Liam; Maloney, Alexander; Perlmutter, Eric: Harmonic analysis of 2d CFT partition functions (2021)
  15. Berger, Tobias; Klosin, Krzysztof: Irreducibility of limits of Galois representations of Saito-Kurokawa type (2021)
  16. Brock, Bradley W.; Elkies, Noam D.; Jordan, Bruce W.: Periodic continued fractions over (S)-integers in number fields and Skolem’s (p)-adic method (2021)
  17. Bruin, Peter; Perucca, Antonella: Reductions of points on algebraic groups. II (2021)
  18. Calegari, Frank; Talebizadeh Sardari, Naser: Vanishing Fourier coefficients of Hecke eigenforms (2021)
  19. Chałupka, Karolina: Cubic forms, powers of primes and classification of elliptic curves (2021)
  20. Dawsey, Madeline Locus; McCarthy, Dermot: Hypergeometric functions over finite fields and modular forms. A survey and new conjectures (2021)

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