Here's the closed form tommy,
\(
\phi(s,a,b,c) = \Omega_{j=1}^\infty e^{a(s-j) + b + cz}\bullet z\\
= \lim_{n\to\infty} e^{\displaystyle a(s-1) + b + ce^{\displaystyle a(s-2) + b + ce^{...a(s-n)+b+cz}}}
\)
This converges for \( \Re(a) > 0, s,b,c \in \mathbb{C} \)--and is holomorphic on these domains; and converges to the same function for all \( z\in\mathbb{C} \).
\(
\phi(s,a,b,c) = \Omega_{j=1}^\infty e^{a(s-j) + b + cz}\bullet z\\
= \lim_{n\to\infty} e^{\displaystyle a(s-1) + b + ce^{\displaystyle a(s-2) + b + ce^{...a(s-n)+b+cz}}}
\)
This converges for \( \Re(a) > 0, s,b,c \in \mathbb{C} \)--and is holomorphic on these domains; and converges to the same function for all \( z\in\mathbb{C} \).