gives the Riemann xi function .


  • Mathematical function, suitable for symbolic and numeric manipulations.
  • xi(s)=1/2 s (s-1) pi^(-s/2) TemplateBox[{s}, Zeta] TemplateBox[{{s, /, 2}}, Gamma].
  • For certain special arguments, RiemannXi automatically evaluates to exact values.
  • RiemannXi is an entire function with no branch cut discontinuities.
  • RiemannXi can be evaluated to arbitrary numerical precision.
  • RiemannXi 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  (23)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values  (4)

Simple exact values are generated automatically:

Value at zero:

Evaluate symbolically:

Find the minimum of RiemannXi[x]:

Visualization  (2)

Plot the RiemannXi:

Plot the real part of the RiemannXi function:

Plot the imaginary part of the RiemannXi function:

Function Properties  (6)

RiemannXi has the mirror property TemplateBox[{TemplateBox[{z}, Conjugate, SyntaxForm -> SuperscriptBox]}, RiemannXi]=TemplateBox[{TemplateBox[{z}, RiemannXi]}, Conjugate]:

RiemannXi is defined through the identity:

RiemannXi threads elementwise over lists:

RiemannXi is neither non-increasing nor non-decreeing:

RiemannXi is not an injective function:

TraditionalForm formatting, while avoiding the evaluation:

Differentiation  (3)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

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

Series Expansions  (4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find the series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Wolfram Research (2014), RiemannXi, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannXi.html.


Wolfram Research (2014), RiemannXi, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannXi.html.


@misc{reference.wolfram_2021_riemannxi, author="Wolfram Research", title="{RiemannXi}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/RiemannXi.html}", note=[Accessed: 27-October-2021 ]}


@online{reference.wolfram_2021_riemannxi, organization={Wolfram Research}, title={RiemannXi}, year={2014}, url={https://reference.wolfram.com/language/ref/RiemannXi.html}, note=[Accessed: 27-October-2021 ]}


Wolfram Language. 2014. "RiemannXi." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RiemannXi.html.


Wolfram Language. (2014). RiemannXi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RiemannXi.html