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Dirichlet Lambda Function

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The Dirichlet lambda function lambda(x) is the Dirichlet L-series defined by

lambda(x)=sum_(n=0)^(infty)1/((2n+1)^x)
(1)
=(1-2^(-x))zeta(x),
(2)

where zeta(x) is the Riemann zeta function. The function is undefined at x=1. It can be computed in closed form where zeta(x) can, that is for even positive n.

The Dirichlet lambda function is implemented in the Wolfram Language as DirichletLambda[x].

It is related to the Riemann zeta function and Dirichlet eta function by

(zeta(nu))/(2^nu)=(lambda(nu))/(2^nu-1)=(eta(nu))/(2^nu-2)
(3)

and

zeta(nu)+eta(nu)=2lambda(nu)
(4)

(Spanier and Oldham 1987). Special values of lambda(n) include

lambda(2)=(pi^2)/8
(5)
lambda(4)=(pi^4)/(96).
(6)

See also

Dirichlet Beta Function, Dirichlet Eta Function, Dirichlet L-Series, Legendre's Chi-Function, Riemann Zeta Function, Zeta Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 807-808, 1972.Spanier, J. and Oldham, K. B. "The Zeta Numbers and Related Functions." Ch. 3 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 25-33, 1987.

Referenced on Wolfram|Alpha

Dirichlet Lambda Function

Cite this as:

Weisstein, Eric W. "Dirichlet Lambda Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletLambdaFunction.html

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