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 *p*th 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,

for .