Dirichlet Lambda Function

$$\lambda(x) \equiv \sum_{n=0}^\infty (2n+1)^{-x} = (1-2^{-x})\zeta(x)$$ (1)

for $x = 2$, 3, ..., 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 $n$. It is related to the Riemann Zeta Function and Dirichlet Eta Function by

$${\zeta(\nu)\over 2^\nu}={\lambda(\nu)\over 2^\nu-1}={\eta(\nu)\over 2^{\nu}-2}$$ (2)

and

$$\zeta(\nu)+\eta(\nu)=2\lambda(\nu)$$ (3)

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

$$\lambda(2) = {\pi^2\over 8}$$ (4)

$$\lambda(4) = {\pi^4\over 96}.$$ (5)

See also Dirichlet Beta Function, Dirichlet Eta Function, Riemann Zeta Function, Zeta Function

References

Abramowitz, M. and Stegun, C. 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.