For a numerical semigroup S ⊆ N with embedding dimension e, conductor c and left part L = S ∩ [0, c − 1], set W (S) = e|L| − c. In 1978 Wilf asked, in equivalent terms, whether W (S) ≥ 0 always holds, a question known since as Wilf's conjecture. Using a closely related lower bound W 0 (S) ≤ W (S), we show that if |L| ≤ 12 then W 0 (S) ≥ 0, thereby settling Wilf's conjecture in this case. This is best possible, since cases are known where |L| = 13 and W 0 (S) = −1. Wilf's conjecture remains open for |L| ≥ 13.