  In this paper, we study contact surgeries along Legendrian links in the
standard contact 3-sphere. On one hand, we use algebraic methods to prove the
vanishing of the contact Ozsv\'{a}th-Szab\'{o} invariant for contact
$(+1)$-surgery along certain Legendrian two-component links. The main tool is a
link surgery formula for Heegaard Floer homology developed by Manolescu and
Ozsv\'{a}th. On the other hand, we use contact-geometric argument to show the
overtwistedness of the contact 3-manifolds obtained by contact $(+1)$-surgeries
along Legendrian two-component links whose two components are linked in some
special configurations.
