I am interested in finding the ordinary generating function of the numerator in the sum
$$\sum_{n = 1}^{\infty}\frac{(-1)^{n+1}}{n^{4m + 3}}$$
I have tried this problem for quite considerable amount of time. After spending some time I was able to figure out that when $m = 0$ then the numerator is $\pi^3$, whereas when $m = 1$ then the numerator turns out to be $13\pi^7$. This seems quite interesting. There's definitely some pattern going on here. However I'm not able to figure it out.
I really don't care much about the denominator since it seems to grow quite large very quickly. I am only interested in the numerator of the infinite series.
Your help would be highly appreciated. Thanks in advance.