Is there a formula for $k\pi ^n$, if $n$ is an odd number and $k$ is a rational number?

Question

I always find the formula for $k\pi ^n$ when $n$ is an even number and $k$ is a rational number, but I did't find for an odd number.

Answer 1

The Dirichlet beta function may satisfy you : $$\beta(x):=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^x}$$ with the table of values : \begin{array} {c|c} n&\beta(n)\\ \hline 1&\frac {\pi}4\\ 2&K\\ 3&\frac {\pi^3}{32}\\ 4&\beta(4)\\ 5&\frac {5\,\pi^5}{1536}\\ \end{array} with $K$ the Catalan constant.

Here the $n$ even cases are the difficult ones as opposed to the $\zeta$ odd cases !

A parallel with $\zeta$ is proposed in this thread (Euler numbers replacing Bernoulli numbers...).

Answer 2

I suspect the formula for $n$ even you are talking about comes from the riemann zeta function seen here. In the same articles you can read that a general formula for $\zeta(2k+1)$ is still an open problem.