I was thinking about how the standard proofs that $\pi$ is irrational use variations on $sin(\pi) =0$, in contrast to Apéry's proofs that $\zeta(2),\zeta(3)$ are irrational (the first of course also proving that $\pi$ is irrational.)

My question is as follows:

Let $$Li_2(x)=\sum_{n\ge1} \frac{x^n}{n^2},$$

and let $f(x)$ denote it's compositional inverse.

What are the complex roots of $f$?

I didn't find anything looking around, and playing with sage makes it look like maybe there are no zeros.

I was thinking about how the standard proofs that $\pi$ is irrational use variations on $sin(\pi) =0$, in contrast to Apéry's proofs that $\zeta(2),\zeta(3)$ are irrational (the first of course also proving that $\pi$ is irrational.)

My question is as follows:

Let $$Li_2(x)=\sum_{n\ge1} \frac{x^n}{n^2},$$

and let $f(x)$ denote its compositional inverse.

What are the complex roots of $f$?

I didn't find anything looking around, and playing with sage makes it look like maybe there are no zeros.