Functional Equations related to the Dirichlet lambda and beta functions

TL;DR

This paper derives explicit formulas for the Dirichlet beta and lambda functions at specific integers, revealing fundamental relations with a new integral-based function J(s).

Contribution

It provides closed-form expressions and fundamental relations for Dirichlet lambda and beta functions at specific integers, based on the new function J(s).

Findings

Closed-form expressions for Dirichlet beta at even integers.

Closed-form expressions for Dirichlet lambda at odd integers.

Fundamental relations between these functions and J(s).

Abstract

We give closed-form expressions for the Dirichlet beta function at even positive integers and for the Dirichlet lambda function at odd positive integers, based on the function J(s) defined via convergent integral. We also show fundamental relations between Dirichlet lambda and beta functions and the function J(s).