I'm young, and have been studying this number for quite some time. Possible suspects for a closed form i have personally encountered through ghetto makeshift studyies are:
- Euler-Mascheroni Constant
- Glaisher Constant
- Cube root of two, i.e $\sqrt[3]2$
- $\displaystyle\frac{\pi\tanh[\pi\sqrt{3}]}{\sqrt{3}}$
- Random values of Inverse Tangent, Inverse Hyperbolic tangent.
The cube root of two and Euler's Constant are especially likely suspects, but I'm confident that the cube root of two is a coefficient for the true closed form. They appear frequently when I'm trying different methods to evaluate $\zeta(3)$.
I would like to hear your opinions, if you have any, about the relationship between known constants and $\zeta(3)$. I know many people believe that odd values of/for $\zeta(3)$ are unique in a sense where they are unrelated to other known constants, but I am hoping this is not true.
Also, I was wondering if someone could help me find the closed form for the real part of a complex valued Digamma function, or if this series is related to $\zeta(3)$ at all.
This is not an answer, but it is too long for a comment.
If you look at sequence $\rm A002117$ at $\rm OEIS$, you will find a very nice approximation of Apéry's constant . It is given by $$\zeta(3) \approx\frac{236 }{197}\log ^3(2)-\frac{283\pi}{394} \log ^2(2)+\frac{11\pi ^2}{394} \log (2)+\frac{209}{394} \log ^3\left(1+\sqrt{2}\right)+\frac{93 \pi C}{197}-\frac{5}{197}$$ and the first $22$ digits are correct.