I thought I'd compile some of the things I know...

[sheldonison]

Hey, Sheldon.

So if I'm reading you correctly, we are trying to look at solutions of;

\( \phi(s)+s=(2k+1)\pi i\\ \)

Then, for \( |x|<\delta \),

\( \phi(s+x+1)+s+1+x=0\\ \)

I updated it to \( \phi(s+x+1) \), but yeah.  The conjecture is that given a suitable definition of these values of s, and what "near" means, then the closest singularities to the real line of \( \ln(\ln(\phi)) \) are near (s+1), and there is a 1:1 correspondence between those singularities and integers k>=0, and that as k increasers these singularities cluster arbitrarily close together, and arbitrarily close to the real axis.

For my calculations, I used Newton's method twice, once to find the values of s for each value of k by starting with the value of s for k-1.  And then I used Newton's method a second time to find the nearby value where \( \phi(s+x+1)+s+1+x=0 \)