gives the Hurwitz zeta function .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Hurwitz zeta function is defined as an analytic continuation of .
  • HurwitzZeta is identical to Zeta for .
  • Unlike Zeta, HurwitzZeta has singularities at for non-negative integers .
  • HurwitzZeta has branch cut discontinuities in the complex plane running from to .
  • For certain special arguments, HurwitzZeta automatically evaluates to exact values.
  • HurwitzZeta can be evaluated to arbitrary numerical precision.
  • HurwitzZeta automatically threads over lists.


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Basic Examples  (6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

Scope  (26)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

Specific Values  (5)

Simple exact values are generated automatically:

HurwitzZeta[s,a] for symbolic a:

HurwitzZeta[s,a] for symbolic s:

Values at zero:

Find a value of s for which HurwitzZeta[s,1]=1.05:

Visualization  (3)

Plot the HurwitzZeta as a function of its parameter s:

Plot the HurwitzZeta function for various orders:

Plot the real part of HurwitzZeta function:

Plot the imaginary part of HurwitzZeta function:

Function Properties  (4)

Real domain of HurwitzZeta for non-negative integers:

Real domain of HurwitzZeta for negative integers:

Complex domain:

Approximate function range of HurwitzZeta:

HurwitzZeta threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to a:

Higher derivatives with respect to a:

Plot the higher derivatives with respect to a when s=3:

Formula for the ^(th) derivative with respect to a:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Function Identities and Simplifications  (2)

HurwitzZeta is defined through the identity:

Recurrence identity:

Properties & Relations  (2)

HurwitzZeta can be generated by symbolic solvers:

For , two-argument Zeta coincides with HurwitzZeta:

Possible Issues  (2)

HurwitzZeta differs from the two-argument form of Zeta by a different choice of branch cut:

HurwitzZeta includes singular terms, unlike Zeta:

Wolfram Research (2008), HurwitzZeta, Wolfram Language function,


Wolfram Research (2008), HurwitzZeta, Wolfram Language function,


@misc{reference.wolfram_2020_hurwitzzeta, author="Wolfram Research", title="{HurwitzZeta}", year="2008", howpublished="\url{}", note=[Accessed: 18-April-2021 ]}


@online{reference.wolfram_2020_hurwitzzeta, organization={Wolfram Research}, title={HurwitzZeta}, year={2008}, url={}, note=[Accessed: 18-April-2021 ]}


Wolfram Language. 2008. "HurwitzZeta." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2008). HurwitzZeta. Wolfram Language & System Documentation Center. Retrieved from