Catalan's constant

Catalan's constant is $$G=\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)^2} = 0.915 965 594 177 219 015 054 603 514 932 384 110 774 \ldots.$$ This means that Catalan's constant can be expressed as $\beta(2)$ where $\beta$ is the Dirichlet beta function.

Properties

Theorem

The following formula holds: $$K=\beta(2),$$ where $K$ is Catalan's constant and $\beta$ denotes the Dirichlet beta function.

References

Catalan's constant using Legendre chi: The following formula holds: $$K=-i\chi_2(i),$$ where $K$ is Catalan's constant and $\chi$ denotes the Legendre chi function. where

Proof: █

Theorem

The following formula holds: $$K=\dfrac{\pi}{24} -\dfrac{\pi}{2}\log(A)+4\pi \zeta' \left(-1 , \dfrac{1}{4} \right),$$ where $K$ is Catalan's constant, $A$ is the Glaisher–Kinkelin constant, and $\zeta'$ denotes the partial derivative of the Hurwitz zeta function with respect to the first argument.