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 32 articles )

Showing results 1 to 20 of 32.
Sorted by year (citations)

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  1. Bruin, Peter; Ferraguti, Andrea: On $L$-functions of quadratic $\mathbbQ$-curves (2018)
  2. Creutz, Brendan: Improved rank bounds from $2$-descent on hyperelliptic Jacobians (2018)
  3. Daniels, Harris B.; Lozano-Robledo, Álvaro; Najman, Filip; Sutherland, Andrew V.: Torsion subgroups of rational elliptic curves over the compositum of all cubic fields (2018)
  4. Fité, Francesc; Guitart, Xavier: Fields of definition of elliptic $k$-curves and the realizability of all genus 2 Sato-Tate groups over a number field (2018)
  5. Kohen, Daniel; Pacetti, Ariel: On Heegner points for primes of additive reduction ramifying in the base field (2018)
  6. Petridis, Yiannis N.; Risager, Morten S.: Arithmetic statistics of modular symbols (2018)
  7. Roberts, David P.: $\mathrmPGL_2(\mathbbF_\ell)$ number fields with rational companion forms (2018)
  8. Bertin, Marie José; Zudilin, Wadim: On the Mahler measure of hyperelliptic families (2017)
  9. Buzzard, Kevin; Lauder, Alan: A computation of modular forms of weight one and small level (2017)
  10. Daniels, Harris B.; Goodwillie, Hannah: On the ranks of elliptic curves with isogenies (2017)
  11. Dummit, David S.; Kisilevsky, Hershy: Decomposition types in minimally tamely ramified extensions of $\mathbb Q$ (2017)
  12. Gordon, Julia; Roe, David: The canonical measure on a reductive $p$-adic group is motivic (2017)
  13. Holmstrom, Andreas; Vik, Torstein: Zeta types and Tannakian symbols as a method for representing mathematical knowledge (2017)
  14. Kim, Kwang-Seob: An example of a $\textPSL_2(\mathbbF_7)$-maximal unramified extension of a quartic number field (2017)
  15. Kühn, Patrick; Robles, Nicolas; Zaharescu, Alexandru: The largest gap between zeros of entire $L$-functions is less than 41.54 (2017)
  16. Savala, Paul: Computing the Laplace eigenvalue and level of Maass cusp forms (2017)
  17. Bartel, Alex; Page, Aurel: Torsion homology and regulators of isospectral manifolds (2016)
  18. Bergeron, Nicolas; Şengün, Mehmet Haluk; Venkatesh, Akshay: Torsion homology growth and cycle complexity of arithmetic manifolds (2016)
  19. Cooper, I. A.; Morris, Patrick W.; Snaith, N. C.: Beyond the excised ensemble: modelling elliptic curve $L$-functions with random matrices (2016)
  20. 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)

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