Sums of infinite series involving the Dirichlet lambda function

The Dirichlet lambda function $\lambda (s)$ is defined for $\mathrm{R}\mathrm{e}(s)>1$ by $$\lambda (s)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{(2n+1{)}^s}\mathrm{.}$$ This function was initially studied by Euler on the real line, where he denoted it by $N(s)$. In this paper, by applying the partial fraction decomposition of $\pi \tan (\pi x)$ and explicit evaluations of the integrals $${\int }_0^{\frac{1}{2}}{x}^{2m-1}\cos (2l\pi x)dx\text{and}{\int }_0^{\frac{1}{2}}{x}^{m-1}\log \cos (\pi x)dx,$$ for positive integers $l$ and $m$, we derive closed-form expressions for several classes of infinite series involving $\lambda (s)$. We also demonstrate that the values $\lambda (k)$ for even integers $k\ge 2$ arise as constant terms in the Fourier expansions of Eisenstein series associated with the congruence subgroup $${\mathrm{\Gamma }}_0(2):=\{\left(\begin{array}{ccc}a& bc& d\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{Z}):c\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}2)\}\mathrm{.}$$