## Experimental Mathematics

- Experiment. Math.
- Volume 13, Issue 2 (2004), 137-150.

### Computing Special Values of Motivic *L*-Functions

#### Abstract

We present an algorithm to compute values *L(s)* and derivatives {\small $L^{(k)}(s)$} of *L*-functions of motivic origin numerically to required accuracy. Specifically, the method applies to any *L*-series whose {\small $\Gamma$}-factor is of the form {\small $A^s\prod_{i=1}^d \Gamma(\frac{s+\lambda_j}{2})$} with *d* arbitrary and complex {\small $\lambda_j$}, not necessarily distinct. The algorithm relies on the known (or conjectural) functional equation for *L(s)*.

#### Article information

**Source**

Experiment. Math., Volume 13, Issue 2 (2004), 137-150.

**Dates**

First available in Project Euclid: 20 July 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1090350929

**Mathematical Reviews number (MathSciNet)**

MR2068888

**Zentralblatt MATH identifier**

1139.11317

**Subjects**

Primary: 11M99: None of the above, but in this section

Secondary: 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture) 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10] 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations

**Keywords**

L-functions Zeta-functions motives Meijer G-function

#### Citation

Dokchitser, Tim. Computing Special Values of Motivic L -Functions. Experiment. Math. 13 (2004), no. 2, 137--150. https://projecteuclid.org/euclid.em/1090350929