I doubt there is an efficient algorithm. Suppose you have a knapsack of size $a_1 - \sum_{i>1} a_i$ and a collection of items each weighing $a_i/2$ for $i > 1$.
Determining if $\mathcal{L}(f_{a_1}f_{a_2}\cdots f_{a_k})$ is non-zero is equivalent to asking if there is a way to pack the knapsack exactly.