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:

Visualize in the complex plane:

The Dirichlet lambda function expands in terms of zeta functions:

Compute some special values:

Scope  (6)

DirichletLambda is not an analytic function:

DirichletLambda has both singularity and discontinuity at x=1:

DirichletLambda is meromorphic:

It has a simple pole at :

DirichletLambda is neither non-decreasing nor non-increasing:

DirichletLambda is not injective:

DirichletLambda is neither non-negative nor non-positive:

DirichletLambda is neither convex nor concave:

Properties & Relations  (1)

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

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_2024_dirichletlambda, author="Wolfram Research", title="{DirichletLambda}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/DirichletLambda.html}", note=[Accessed: 23-July-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_dirichletlambda, organization={Wolfram Research}, title={DirichletLambda}, year={2014}, url={https://reference.wolfram.com/language/ref/DirichletLambda.html}, note=[Accessed: 23-July-2024 ]}