Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 5

Theorem

Let $\beta$ denote the Dirichlet beta function.

Then:

$\map \beta 5 = \dfrac {5 \pi^5} {1536} $

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

\(\ds \leadsto \ \ \)

\(\ds \map \beta 5\)

\(=\)

\(\ds \paren {-1}^2 \dfrac {E_4 \pi^5 } {4^3 \paren {4}!}\)

setting $n := 2$

\(\ds \)

\(=\)

\(\ds \frac {5 \pi^5} {1536}\)

Euler Number Values: $E_4 = 5$

$\blacksquare$