# Overview

A large variety of $\zeta$-functions and $L$-functions occur in number theory and algebraic geometry. Some well-known $L$-functions include the Riemann $\zeta$-function, the Dedekind $\zeta$-function of a number field, Dirichlet series associated to characters, and $L$-series of curves (e.g., elliptic curves) over the rationals. Magma provides functionality for constructing such $L$-functions and computing their values in the complex plane. A typical calculation might go as follows:

```
> L := LSeries(EllipticCurve([0, -1, 1, 0, 0]));
> Evaluate(L,2);
0.546048036215013518334126660433

```

The first line defines an $L$-series $L(E,s)$ of the elliptic curve

$$
E: y^2+xy=x^3-x^2
$$

while the second line computes its value at $s=2$. An impatient reader may wish simply to type `LSeries;` at the prompt and look at the various `LSeries` signatures and mimic the code above, thus getting access to much of the functionality.

Topics covered in this chapter include:

- The built-in $L$-series which include the Riemann $\zeta$-function, the Dedekind $\zeta$-function of a number field, Dirichlet series associated to characters, Artin representations, modular forms, and $L$-series of elliptic curves;

- The calculation of values, derivatives and Taylor expansions of $L$-series at a complex point $s_0$ to desired accuracy;

- A technical description of the $L$-series object in Magma, together with a description of how to construct user-defined $L$-series with any number of gamma factors, provided that the $L$-series satisfies a functional equation of the standard type;

- Operations such as division, multiplication and the tensor product of two $L$-series.

The reader is referred to Manin-Panchishkin [[Shafarevich, 1995](../../references.md#cite-emsnt1)] Chapter 4, Serre [[Serre, 1965](../../references.md#cite-serre63)] and articles in [[Jannsen *et al.*, 1994](../../references.md#cite-motives94)] for a background on $L$-functions. The algorithms mostly follow Dokchitser [[Dokchitser, 2004](../../references.md#cite-dok04)] and the Pari implementation ComputeL [[Dokchitser, 2002](../../references.md#cite-dokcomputel)]. See also Lavrik [[Lavrik, 1967](../../references.md#cite-lavrik67)], Tollis [[Tollis, 1997](../../references.md#cite-tollis97)] and the exposition in Cohen [[Cohen, 2000](../../references.md#cite-cohen2)], 10.3.