In Mathworld’s entry on the Riemann zeta function, one finds in eq. 119-121 the curious evaluations,
However, using the Inverse Symbolic Calculator, the first and the third, plus another one, can also be expressed as,
where . Interesting similar forms, isn’t it?
Unfortunately, it doesn’t seem to generalize to for p = 8. However, there is still p = 3 and, based on the even case, I assumed perhaps roots of unity are also involved. First, given the Euler-Mascheroni constant , and the digamma function,
where we suppress the subscript for ease of notation. Define,
and the pth root chosen such that , then I found that p = 3 generalizes as,
and so on, though a rigorous proof is needed that it holds true for all odd numbers p.
P.S. Going back to even p, note that p = 2, 4, 6 can also be expressed by the digamma function since,