  We can associate local constant to every continuous finite dimensional
complex representation of the absolute Galois group $G_F$ of a non-archimedean
local field $F/\mathbb{Q}_p$ by Deligne and Langlands. To give explicit formula
of local constant of a representation, we need to compute $\lambda$-functions
explicitly. In this thesis we compute $\lambda_{K/F}$ explicitly, where $K/F$
is a finite degree Galois extension of a non-archimedean local field $F$,
except when $K/F$ is a wildly ramified quadratic extension with
$F\ne\mathbb{Q}_2$.
  Then by using this $\lambda$-function computation, in general, we give an
invariant formula of local constant of finite dimensional Heisenberg
representations of the absolute Galois group $G_F$ of a non-archimedean local
field $F$. But for explicit invariant formula of local constant for a
Heisenberg representation, we should have information about the dimension of a
Heisenberg representation and the arithmetic description of the determinant of
a Heisenberg representation. In this thesis, we give explicit arithmetic
description of the determinant of Heisenberg representation.
  We also construct all Heisenberg representations of dimensions prime to $p$,
and study their various properties. By using $\lambda$-function computation and
arithmetic description of the determinant of Heisenberg representations, we
give an invariant formula of local constant for a Heisenberg representation of
dimension prime to $p$.
