  We consider symmetric (not complete intersection) numerical semigroups S_6,
generated by a set of six positive integers {d_1,...,d_6}, gcd(d_1,...,d_6)=1,
and derive inequalities for degrees of syzygies of such semigroups and find the
lower bound for their Frobenius numbers. We show that this bound may be
strengthened if S_6 satisfies the Watanabe lemma.
