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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I'd say no, see particular values of polylogarithm (at $z= \pm 1$) – reuns Sep 23 '17 at 07:23
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I don't know about $\zeta(s)$, but the logarithmic derivative $\frac{\zeta'}{\zeta}(s)=\frac{\zeta'(s)}{\zeta(s)}$ has a closed-form representation at least at one rational non-integer point which is $\frac{\zeta'}{\zeta}\left(\frac{1}{2}\right)=\frac{1}{2}\left(\frac{\pi}{2}+\gamma+\log (8 \pi)\right)$. – Steven Clark Feb 25 '20 at 22:20
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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
