Using ladder operators to find eigenfunctions

Question

Using $L_1\pm iL_2$ as raising and lowering operators I need to obtain an expression for $Y_{l,l}(\theta,\phi)$ and after this I need to find $Y_{l,m}(\theta,\phi)$ for $(l,m)=(1,1),(1,0),(1,-1),(2,2),(2,1)$ and $(2,0)$ The hint says that $Y_{l,l}(\theta,\phi)$, $Y_{1,1}(\theta,\phi)$, $Y_{2,2}(\theta,\phi)$ should be easy to find as they are in the kernel of $L_+$ and are eigenfunctions of $L_3$. I think I know how to 'lower' these to find the rest but I am struggling to find these. I got the stage where I have $Y_{l,m}(\theta,\phi) = P_{l,m}(\theta)e^{im\phi}$ and I'm not sure where to go from here, or if this is even correct! Any help will be appreciated!