In this paper we deal with generic expansions of first-order predicate logics of some left-continuous t-norms with a countable set of truth-constants. Besides already known results for the case of Lukasiewicz logic, we obtain new conservativeness and completeness results for some other expansions. Namely, we prove that the expansions of predicate Product, Gödel and Nilpotent Minimum logics with truth-constants are conservative, which already implies the failure of standard completeness for the case of Product logic. In contrast, the expansions of predicate Gödel and Nilpotent Minimum logics are proved to be strong standard complete but, when the semantics is restricted to the canonical algebra, they are proved to be complete only for tautologies. Moreover, when the language is restricted to evaluated formulae we prove canonical completeness for deductions from finite sets of premises.