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.

Evaluate the prime zeta function:

Find the integral representations of the Dirichlet eta function: