What is known about the complex solutions to $\zeta(s)=-1$?

Question

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.

Answer 1

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.