  The standard $L$-functions of $\mathrm{GL}_{2n}$ expressed in terms of the
Friedberg-Jacquet global zeta integrals have better structure for arithmetic
applications, due to the relation of the linear periods with the modular
symbols. The most technical obstacles towards such arithmetic applications are
(1) non-vanishing of modular symbols at infinity and (2) the existance or
construction of uniform cohomological test vectors. Problem (1) is also called
the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by
establishing the existence of certain cohomological test vectors.
  In this paper, we explicitly construct an archimedean local integral that
produces a new type of a twisted linear functional $\Lambda_{s,\chi}$, which,
when evaluated with our explicitly constructed cohomological vector, is equal
to the local twisted standard $L$-function $L(s,\pi\otimes\chi)$ as a
meromorphic function of $s\in \mathbb{C}$. With the relations between linear
models and Shalika models, we establish (1) with an explicitly constructed
cohomological vector, and hence recovers a non-vanishing result of Binyong Sun
via a completely different method. Our main result indicates a complete
solution to (2), which will be presented in a paper of Dihua Jiang, Binyong Sun
and Fangyang Tian with full details and with applications to the global period
relations for the twisted standard $L$-functions at critical places.
