Let
F
F
be a non-Archimedean local field and
n
1
n_{1}
,
n
2
n_{2}
positive integers. For
i
=
1
,
2
i=1,2
, let
G
i
=
G
L
n
i
(
F
)
G_{i}=\mathrm {GL}_{n_{i}}(F)
and let
π
i
\pi _{i}
be an irreducible supercuspidal representation of
G
i
G_{i}
. Jacquet, Piatetskii-Shapiro and Shalika have defined a local constant
ε
(
π
1
×
π
2
,
s
,
ψ
)
\varepsilon (\pi _{1}\times \pi _{2},s,\psi )
to the
π
i
\pi _{i}
and an additive character
ψ
\psi
of
F
F
. This object is of central importance in the study of the local Langlands conjecture. It takes the form
ε
(
π
1
×
π
2
,
s
,
ψ
)
=
q
−
f
s
ε
(
π
1
×
π
2
,
0
,
ψ
)
,
\begin{equation*}\varepsilon (\pi _{1}\times \pi _{2},s,\psi ) = q^{-fs}\varepsilon (\pi _{1} \times \pi _{2},0,\psi ), \end{equation*}
where
f
=
f
(
π
1
×
π
2
,
ψ
)
f=f(\pi _{1}\times \pi _{2},\psi )
is an integer. The irreducible supercuspidal representations of
G
=
G
L
n
(
F
)
G=\mathrm {GL}_{n}(F)
have been described explicitly by Bushnell and Kutzko, via induction from open, compact mod centre, subgroups of
G
G
. This paper gives an explicit formula for
f
(
π
1
×
π
2
,
ψ
)
f(\pi _{1} \times \pi _{2},\psi )
in terms of the inducing data for the
π
i
\pi _{i}
. It uses, on the one hand, the alternative approach to the local constant due to Shahidi, and, on the other, the general theory of types along with powerful existence theorems for types in
G
L
(
n
)
\mathrm {GL}(n)
, developed by Bushnell and Kutzko.