Could someone please evaluate the integral: $$P(\vec{l};z)=\frac{1}{(2\pi)^2}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\frac{\text{exp}(-i(l_{1}k_{1}+l_{2}k_{2})d{k_{1}}d{k_{2}}}{1-z^2\lambda(\vec{k})}$$ $\vec{l}$ is the direction variable. The integral is a representative of a random walk on the honeycomb lattice.
$$\lambda(k)=\frac{1}{9}[1+(2\cos(k_{1}))^2+4\cos(k_{1})\cos(k_{2})]$$
$k_{1}$ and $k_{2}$ are the Fourier variables and $z$ is the $z$ transform/generating function variable.
Edit source for the integral: https://arxiv.org/pdf/1004.1435.pdf
Edit: I made some progress in the equation post modifying the integral to account for the basis vectors,
We can take $\exp(-ik_1)=t_1$ and similarly for $k_{2}$, then we can write the cosine terms in terms of $t_{1}$ and $t_{2}$ which we assume are complex valued and hence we can evaluate it using contour integrals?
Edit2Another approach that I am exploring is expanding the terms-
