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I would like to know do we have closed-form of Riemann zeta at at least one rational non-integer point such that that closed form contains already known constants and is not an infinite sum?

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  • $\begingroup$ I'd say no, see particular values of polylogarithm (at $z= \pm 1$) $\endgroup$ – reuns Sep 23 '17 at 7:23
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You can create approximations as hyperbolas:

$\zeta(x)=\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^x} \rightarrow H(x)=\frac{\alpha x+\beta}{\lambda x-x_0}+y_0$

One for x>1 and a different for x<1. To fit your formula you need some exist values.

For samples and more details please read my feedback here

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