Examples for

# Zeta Functions

Zeta functions are a family of special functions defined through a Dirichlet series. The most famous example is the Riemann zeta function. Wolfram|Alpha can compute values for multiple variants of zeta functions as well as help you explore other functionalities, such as visualization and series expansion.

### Riemann Zeta Function

Compute the properties for the Riemann zeta function or get a Riemann zeta function calculator.

#### Compute the exact values of the zeta function:

#### Numerically evaluate zeta near a nontrivial zero:

#### Plot the zeta function on the critical line:

#### Find series representations of the zeta function:

#### Find a functional equation for the zeta function:

### Riemann–Siegel Functions

Visualize or compute the Riemann–Siegel Z function and the Riemann–Siegel theta function.

#### Plot the Riemann–Siegel functions:

### Hurwitz Functions

Generalize the Riemann zeta function to the Hurwitz zeta function and further to the Lerch transcendent.

#### Analyze a series for the Hurwitz zeta function:

#### Plot the Hurwitz–Lerch transcendent:

### Other Zeta Functions

Visualize or compute other zeta-family functions.