Fast approximation of $\sum_{i=1}^n e^{-\beta \ln(i)+\gamma{\ln}^2(i)}$ for large $n$

Question

I need a numerical approximation with low computational complexity of

$$\sum_{i=1}^n e^{-\beta \ln(i)+\gamma{\ln}^2(i)}$$ for $n\approx10^6$, where $1\lt\beta\lt3$ and $0<\gamma<0.05$

I have unsuccessfully (and naively) tried to use the Euler–Maclaurin formula. Would anyone have a suggestion ?