given the series $$ B (s)=\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n-1)^s} $$
is there a realtion (functional relation) with other series like Riemann zeta function
or $$ \lambda (s)= \sum_{n=1}^{\infty}\frac{1}{(2n-1)^s}$$
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2Let $\chi_4(2n)=0,\chi_4(2n+1)=(-1)^n$, then $B(s) = L(s,\chi_4)$. The Dirichlet L-functions have similar properties to $\zeta(s)$ but they are all distinct and don't have any simple relation with each other. – reuns Dec 16 '22 at 11:42
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1You may try $\sum a_{2n}+\sum a_{2n-1}=\sum a_n$ – Тyma Gaidash Dec 16 '22 at 13:10
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1$\beta (s)=\sum\limits_{n=0}^\infty \frac{(-1)^n}{(2 n+1)^s}=\sum\limits_{n=1}^\infty \frac{(-1)^{n-1}}{(2 n-1)^s}=4^{-s} \left(\zeta \left(s,\frac{1}{4}\right)-\zeta \left(s,\frac{3}{4}\right)\right)$, whereas $\lambda(s)=\sum\limits_{n=1}^\infty \frac{1}{(2 n-1)^s}=\left(1-2^{-s}\right) \zeta (s)$. – Steven Clark Dec 16 '22 at 15:38