Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 3

Theorem

Let $\beta$ denote the Dirichlet beta function.

Then:

$\map \beta 3 = \dfrac {\pi^3} {32} $

Proof

\(\ds \map \beta {2 n + 1}\)

\(=\)

\(\ds \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}\)

Dirichlet Beta Function at Odd Positive Integers

\(\ds \leadsto \ \ \)

\(\ds \map \beta 3\)

\(=\)

\(\ds \paren {-1}^1 \dfrac {E_2 \pi^3 } {4^2 \paren {2}!}\)

setting $n := 1$

\(\ds \)

\(=\)

\(\ds \frac {\pi^3} {32}\)

Euler Number Values: $E_2 = -1$

$\blacksquare$