On numerical semigroups with at most 12 left elements

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