Any possible suspects for $\zeta(3)$? 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.
 A: 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.
A: No closed-form expression is known for $\zeta(3)$. The closest you can get to an expresison involing known constants are due to Plouffe and Borwein & Bradley:
$$
\begin{aligned}
\zeta(3)&=\frac{7\pi^3}{180}-2\sum_{n=1}^\infty \frac{1}{n^3(e^{2\pi n}-1)},\\
\sum_{n=1}^\infty \frac{1}{n^3\,\binom {2n}n} &= -\frac{4}{3}\,\zeta(3)+\frac{\pi\sqrt{3}}{2\cdot 3^2}\,\left(\zeta(2, \tfrac{1}{3})-\zeta(2,\tfrac{2}{3}) \right).
\end{aligned}
$$
Moreover, in this Math.SE post we have:
$$
\frac{3}{2}\,\zeta(3) = \frac{\pi^3}{24}\sqrt{2}-2\sum_{k=1}^\infty \frac{1}{k^3(e^{\pi k\sqrt{2}}-1)}-\sum_{k=1}^\infty\frac{1}{k^3(e^{2\pi k\sqrt{2}}-1)}.
$$
You can also check out this paper by Vepstas, which provides a nice generalization to some of these identities.
