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Context
At the bottom of this page from Wolfram Mathworld on the Riemann zeta function, the following constants are defined in equations (133), (137), and (141), respectively:

\begin{align*} C_{1} &:= \sum_{n=2}^{\infty} \frac{\zeta(n)}{n!} \\ & \approx 1.078188729575818482758265436769832381707219, \\\qquad C_{2} &:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{n!} \\ & \approx 2.407446554790328514709486656223022725582266, \text{ and} \\ C_{3} &:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!} \\ &\approx 0.869001991962908998811054805561395688892494.\end{align*} It is also stated that "These sums have no known closed-form expression".

However, it could be the case that there are conjectured closed forms for these constants that correspond to the actual values with a high degree of accuracy.

Question: Are there any of such conjectured closed forms for $C_{1}$, $C_{2}$, and $C_{3}$ ?

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    $\begingroup$ For the first one I conjecture $e^{-31/48-23/(48e)+e/24-5/(24\pi)-7\pi/48}\pi^{23e/48-1/6}\sqrt[24]{-\tan(\pi e)}=1.078188729575818482\dots$ :) $\endgroup$ Aug 9, 2022 at 23:26
  • $\begingroup$ $$C_1 = \int_0^{ + \infty } {\frac{1}{{e^x - 1}}\left[ {\frac{{I_1 (2\sqrt x )}}{{\sqrt x }} - 1} \right]dx}$$ $\endgroup$
    – Gary
    Aug 10, 2022 at 1:10
  • $\begingroup$ @Gary Indeed, such formulas are also stated on the Mathworld page (see eq. 134) $\endgroup$
    – Max Muller
    Aug 10, 2022 at 12:51
  • $\begingroup$ @MaxMuller Oh, I should have checked it. $\endgroup$
    – Gary
    Aug 10, 2022 at 13:51

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