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DirichletLambda[s]

gives the Dirichlet lambda function TemplateBox[{s}, DirichletLambda].

Details
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Examples  
Basic Examples  
Scope  
Properties & Relations  
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DirichletLambda

DirichletLambda[s]

gives the Dirichlet lambda function TemplateBox[{s}, DirichletLambda].

Details

  • Mathematical function, suitable for both symbolic and numeric manipulation.
  • For , the Dirichlet lambda function is defined as TemplateBox[{s}, DirichletLambda]=sum_(n=0)^infty1/((2n+1)^s).
  • For certain special arguments, DirichletLambda automatically evaluates to exact values.
  • DirichletLambda has no branch cut discontinuities.
  • DirichletLambda can be evaluated to arbitrary numerical precision.
  • DirichletLambda automatically threads over lists.

Examples

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Basic Examples  (4)

Plot on the real axis:

Wolfram Language code: Plot[DirichletLambda[s], {s, -10, 5}]

Visualize in the complex plane:

Wolfram Language code: ComplexPlot3D[DirichletLambda[z], {z, 5}]

The Dirichlet lambda function expands in terms of zeta functions:

Wolfram Language code: DirichletLambda[s]

Compute some special values:

Wolfram Language code: DirichletLambda /@ {2, 4}

Scope  (8)

DirichletLambda is not an analytic function:

Wolfram Language code: FunctionAnalytic[DirichletLambda[x], x]

DirichletLambda has both singularity and discontinuity at x=1:

Wolfram Language code: FunctionSingularities[DirichletLambda[x], x]
Wolfram Language code: FunctionDiscontinuities[DirichletLambda[x], x]

DirichletLambda is meromorphic:

Wolfram Language code: FunctionMeromorphic[DirichletLambda[x], x]

It has a simple pole at :

Wolfram Language code: FunctionPoles[DirichletLambda[x], x]

DirichletLambda is neither non-decreasing nor non-increasing:

Wolfram Language code: FunctionMonotonicity[DirichletLambda[x], x]

DirichletLambda is not injective:

Wolfram Language code: FunctionInjective[DirichletLambda[x], x]
Wolfram Language code: Plot[{DirichletLambda[x], .05}, {x, -3, 2}]

DirichletLambda is neither non-negative nor non-positive:

Wolfram Language code: FunctionSign[DirichletLambda[x], x]

DirichletLambda is neither convex nor concave:

Wolfram Language code: FunctionConvexity[DirichletLambda[x], x]

Compute average-case statistical intervals using Around:

Wolfram Language code: DirichletLambda[ Around[2, 0.01]]

Compute the elementwise values of an array:

Wolfram Language code: DirichletLambda[{{1 / 2, -1}, {0, 1 / 2}}]

Or compute the matrix DirichletLambda function using MatrixFunction:

Wolfram Language code: MatrixFunction[DirichletLambda, {{1 / 2, -1}, {0, 1 / 2}}]//FullSimplify

Properties & Relations  (1)

Verify the interrelationship among the DirichletLambda, DirichletEta and Zeta functions:

Wolfram Language code: (Zeta[s]/2^s) == (DirichletLambda[s]/2^s - 1) == (DirichletEta[s]/2^s - 2)//FullSimplify
Wolfram Research (2014), DirichletLambda, Wolfram Language function, https://reference.wolfram.com/language/ref/DirichletLambda.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_dirichletlambda, author="Wolfram Research", title="{DirichletLambda}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/DirichletLambda.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_dirichletlambda, organization={Wolfram Research}, title={DirichletLambda}, year={2014}, url={https://reference.wolfram.com/language/ref/DirichletLambda.html}, note=[Accessed: 13-June-2026]}