About a proof of the functional equation of the Dedekind eta function

According to the book Number Theory II: Iwasawa Theory and Automorphic Forms by Nobushige Kurokawa, Masato Kurihara and Takeshi Saito, one way to prove the functional equation of the Dedekind eta function $\eta(-\frac{1}{z})=\sqrt{-iz}\eta(z)$ is as follows:

Consider this :
$$\log\eta(z)=\log(q^\frac{1}{24})+\sum_{n=1}^{\infty}c(n)q^n$$ Since that satisfies $\sum_{n=1}^{\infty}\frac{c(n)}{n^s}=-\zeta(s)\zeta(s+1)$, using the symmetry between $s$ and $-s$ will yield a proof.

They did not give the entire process. Here is my attempt for that :

Let :
$$F(z)=\sum_{n=1}^{\infty}\log(1-e^{2\pi inz})=\sum_{n=1}^{\infty}c(n)e^{2\pi inz}$$ $$\Phi(s)=\int_0^{\infty}F(iy)y^{s-1}dy$$

Now with the functional equation of the Riemann zeta function, I found $\Phi(s)=\Phi(-s)$, and by a change of variable, I concluded that $F(iy)=F(\frac{i}{y})$, though this is incompatible with the functional equation of the Dedekind eta function.
So what is wrong? It might be the convergence issues, but how to prove it with this method?