The Dirichlet lambda function $\lambda(s)$ is defined for $\mathrm{ Re}(s) > 1$ by \[ \lambda(s) = \sum_{ n=0}^{ \infty} \frac{ 1}{ (2n+1)^s}. \] 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 \[ \int_0^{ \frac{ 1}{ 2}} x^{ 2m-1} \cos(2l\pi x) dx \quad \text{ and} \quad \int_0^{ \frac{ 1}{ 2}} x^{ m-1} \log \cos(\pi x) dx, \] 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 $k \geq 2$ arise as constant terms in the Fourier expansions of Eisenstein series associated with the congruence subgroup \[ \Gamma_0(2) := \left\{ \begin{ pmatrix} a & b c & d \end{ pmatrix} \in \operatorname{ SL}_2(\mathbb{ Z}) : c \equiv 0 \pmod{ 2} \right\}. \]
Categories: math.NT math-ph math.CA math.MP