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I don't recall ever coming across a discussion of the complex solutions to the equation $$\zeta(s)=-1,$$ where $s\in\mathbb{C}$. How many such solutions exist? Is there any literature on this?

Matematica gives a real numerical solution $$s=0.3453726572911539895391917441487543036...$$

References 1, 2 courtesy of Sary's answer.

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  • $\begingroup$ If we were able to give an exact answer to this question we should be able to solve $\zeta(s)=0$ and the Riemann hypothesis, too. $\endgroup$ – Jack D'Aurizio Feb 19 '15 at 14:44
  • $\begingroup$ My thoughts too. I was just wondering if there was anything in the literature on this "other case". Are there multiple solutions or just one solution, etc. For example, in the case $\zeta(s)=0$ it is known that there are an infinite number of such $s$ on the critical line. NB I've changed the title to reflect what I'm trying to get at. $\endgroup$ – Pixel Feb 19 '15 at 14:46
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These are so-called "$a$-points" of the $\zeta$ function. There's a recent preprint of S. Lester containing references to other works in that subject, notably that paper of Levinson.

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    $\begingroup$ I suppose this is related to "level curves" of a function... $\endgroup$ – Pixel Feb 19 '15 at 16:14
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    $\begingroup$ Yes except here the curves are points :) It's also an interesting question to look at $|\zeta(s)|=a$ for positive a (which would give you curves in the plane). I'm not sure what's been done there. $\endgroup$ – Sary Feb 19 '15 at 16:25

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