How to train ML algorithm with multiple values in target data?
Because your test cases may have values from L3 as input, this strikes me as more of a recommender system; consider encoding each possible value in L3 as a new binary column (thinking of them as products that each row/user has purchased / expressed interest in), and look into different kinds of recommender systems to see what seems most appropriate for your data.

(I'm not too familiar with recommenders.  AIUI, these are all clustering algorithms, but with different notions of distance and focusing on either the products or the users.  sklearn does not appear to have any built-ins specifically for recommendations, but you could use it for an underlying model.  There are also some other Python packages specifically for recommenders, but I can't recommend any [see what I did there?].)