What are all the roots of this function modified from Zeta:
$$ F(s)= \dfrac{1}{\pi^s}+ \dfrac{1}{(2\pi)^s}+ \dfrac{1}{(3\pi)^ s} +\dfrac{1}{(4\pi)^s}+ \dfrac{1}{(5\pi)^ s} + ..? $$
Does it converge? Please give links to known references. I am relatively new to nt.
Your series is
$$\begin{align*} F(s) &=\dfrac{1}{\pi^s}+ \dfrac{1}{(2\pi)^s}+ \dfrac{1}{(3\pi)^ s} +\dfrac{1}{(4\pi)^s}+ \dfrac{1}{(5\pi)^ s} + \cdots\\ &= \frac{1}{\pi^s}\left(1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+\cdots \right )\\ &=\frac{\zeta(s)}{\pi^s}. \end{align*}$$
Since your series only converges for $\Re[s]>1$, and $1/\pi^s$ and $\zeta(s)$ have no zeros for $\Re[s]>1$, it follows that your series has no roots.
As for the roots of the analytic continuation of your series, the roots of this would be exactly those of the analytic continuation of the Dirichlet series for $\zeta(s)$ because $1/\pi^s$ has no roots or poles.
You can read more about the convergence of Dirichlet series here, in particular, the section starting on page 23 may be of interest to you.