Is $\sum_{n=0}^\infty (-1)^n (2n+1)^{-s}$ expressible in terms of the zeta function?

Question

I am interested if the alternating Dirichlet Lambda function $$\sum_{n=0}^\infty (-1)^n (2n+1)^{-s}$$

can be expressed in terms of the zeta function or another Dirichlet L-Function.

I know that for $s=1$, the series is equal to the Leibniz formula for $\frac{\pi}{4}$. However, I cannot identify any other values or relationships of this series. Any help is appreciated.

Answer 1

The MathWorld article Dirichlet Beta function gives $\;\beta(x) := \sum_{n=0}^\infty (-1)^n (2n+1)^{-s}\;$ as the definition. It mentions $\;\beta(x) = 4^{-x}(\zeta(x,1/4)-\zeta(x,3/4)),\;$ using the Hurwitz zeta function.