  Let $\pi$ be an irreducible admissible (complex) representation of $GL(2)$
over a non-archimedean characteristic zero local field with odd residual
characteristic. In this paper we prove the equality between the local symmetric
square $L$-function associated to $\pi$ arising from integral representations
and the corresponding Artin $L$-function for its Langlands parameter through
the local Langlands correspondence. With this in hand, we show the stability of
local symmetric $\gamma$-factors attached to $\pi$ under highly ramified
twists.
