This list of mythical creatures that begin with this letter and I am constantly gathering information from each continent, and is continually being updated.
Laelaps (Greek)
Laelaps was a magical dog which was destined to always catch its quarry. It was first given to Europa of Krete (Crete) by Zeus and then passed down to Minos, Prokris (Procris), and finally the hero Kephalos (Cephalus). There are no stories concerning Laelaps parentage, but there are some who suggest that it is the same dog that watched over baby Zeus on Crete.
Kephalos set the hound on the trail of the Teumessian Fox, a monstrous beast ravaging the countryside of Thebes. However, the fox was destined never to be caught, and so Zeus--faced with the paradox of an uncatchable fox being chased by an inescapable hound--turned the pair to stone.