Q:

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

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. 

A:

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...).

