If $\displaystyle L(z)=\frac {az+b}{cz+d}$ where $\displaystyle a, b, c, d \in \mathbb{R}$ and $\displaystyle ad-bc=1$

Show that:

$\displaystyle Im(L(z))=\frac {Im(z)}{|cz+d|^2}$ where $\displaystyle Im(z)=\frac {z-z^{*}}{2i}$.

I think that $\displaystyle Im(L(z))=\frac {L(z)-L(z)^{*}}{2i}$ can be written as the above statement but I am stuck on how to get the conjugate of L(z). I am a little rusty with complex variables. Thanks for your help.