The Riemann zeta function is the prototypical L-function. It is the only L-function of degree 1 and conductor 1, and (conjecturally) it is the only primitive L-function with a pole.
Dirichlet series
| $\zeta(s)$ = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
Functional equation
\[\begin{aligned}\xi(s)=\mathstrut &\Gamma_{\R}(s) \, \zeta(s)\cr=\mathstrut & \,\xi(1-s)\end{aligned}\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(1\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | \((1,\ 1,\ (0:\ ),\ 1)\) |
Euler product
\[\begin{aligned}\zeta(s) = \prod_p (1 - p^{-s})^{-1}\end{aligned}\]
Particular Values
\[\zeta(1/2) \approx -1.460354508\]
Pole at \(s=1\)