Closed form of, or series for $\int_{\epsilon-i\infty}^{\epsilon+i\infty}\frac{e^{az+b^2z^2}}{\sin\pi z}\,dz$

Question

I've been trying to find a closed form expression/series expansion for the following integral without success:

$$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)\,dz=\pi\int_{\epsilon-i\infty}^{\epsilon+i\infty}\frac{e^{az+b^2z^2}}{\sin\pi z}\,dz$$

for some $\epsilon\in(0,1)$. Any input is greatly appreciated!