Can anyone tell me what the Ranicki symmetric Lgroups $L^*(F)$ are when $F$ is a finite field? (and maybe provide a reference?) Thanks!

The symmetric $L$group $L^*(F)$ of a field $F$ are 4periodic, $$L^n(F)=L^{n+4}(F)$$ by Proposition 7.1 of http://www.maths.ed.ac.uk/~aar/papers/ats1.pdf $L^{2i}(F)$ is the Witt group of $()^i$symmetric forms: see Milnor and Husemoller! $L^{2i+1}(F)=0$, see http://www.maths.ed.ac.uk/~aar/papers/simple.pdf (my shortest paper). 

