The Gregory function and its completed Gregory transform

We study the entire interpolation $$\mathcal{G}(z)=\int_0^1 \binom{x}{z}\,dx$$ of the Gregory coefficients. Its completion satisfies the positive Markov-transform identity $$\frac{\pi z}{\sin(\pi z)}\mathcal{G}(z) =\sum_{n=1}^{\infty}\frac{n\left|G_n\right|}{n-z}.$$ Consequently, every zero is real and simple; the negative zeros are the integers $-1,-2,\ldots$, and one zero $\rho_n$ lies in each $(n,n+1)$. We derive complete logarithmic asymptotics for $\rho_n-n$, determine the Cartwright growth and canonical products of $\mathcal{G}$, and realize $1/\rho_n$ spectrally. The resulting relative determinant yields $$\gamma=\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{\rho_n}\right).$$