Another useful way to study local time is related to the very useful *occupation formula* (its meaning is obvious if you think a little bit): $\int_0^t g(W_s) ds = \int_{\mathbb R} g(x) L(x,t) dx $. 

Putting $g(x) = e^{izx}$:
$$
\int_{\mathbb R} e^{izx} L(x,t) dx = \int_0^t e^{iz W_s}ds.
$$
Now the lhs is the Fourier transform of $L$; inverting it, we get
$$
L(x,t) = \frac{1}{2\pi} \int_{\mathbb R} e^{-izx} \int_0^t e^{iz W_s} ds\\, dz = \frac{1}{2\pi} \int_{\mathbb R} \int_0^t e^{iz (W_s-x)} ds\\, dz.
$$
From here one can e.g. more or less easily find the moments of $L$.