I doubt there is an efficient algorithm. Suppose you have a knapsack of size $\sum_{i} a_i$ and a collection of items each weighing $2a_i$ 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.