Let
π
i
\pi _i
be an irreducible cuspidal automorphic representation of
G
L
2
\mathrm {GL}_2
with central character
ω
i
\omega _i
. When
ω
1
ω
2
ω
3
\omega _1\omega _2\omega _3
is trivial, Atsushi Ichino proved a formula for the central value
L
(
1
2
,
π
1
×
π
2
×
π
3
)
L(\frac {1}{2}, \pi _1\times \pi _2\times \pi _3)
of the triple product
L
L
-series in terms of global trilinear forms. We will extend this formula to the case when
ω
1
ω
2
ω
3
\omega _1\omega _2\omega _3
is a quadratic character, giving a non-vanishing criterion of a local trilinear form in terms of the central value of the gamma factor.