Searching for an asymptotic to exp[0.5]

[sheldonison]

All of the zeros of the asymptotic exp[0.5] are roughly in correspondence with -exp(hn), on the negative part of the real axis. A better approximation might be the real crossings of the branch of the Kneser half exponential, that we usually don't look at, at the negative real axis, where this branch is arrived by by circling counter clockwise around L, and this branch is not real valued at the real axis.

So, anyway, lets say we have these zeros of the asymptotic half iterate in an infinite list. Can we recover the asymptotic half iterate with the Weiestrass factorization theorem?
- Sheldon