  Studying the analytic properties of the partial Langlands $L$-function via
Rankin-Selberg method has been proved to be successful in various cases. Yet in
few cases is the local theory studied at the archimedean places, which causes a
tremendous gap to complete the analytic theory of the complete $L$-function. In
this paper, we will establish the meromorphic continuation and the functional
equation of the archimedean local integrals associated with D. Ginzburg's
global integral for the adjoint representation of $\mathrm{GL}_3$. Via the
local functional equation, the local gamma factor
$\Gamma(s,\pi,\mathrm{Ad},\psi)$ can be defined. In a forthcoming paper, we
will compute the local gamma factor $\Gamma(s,\pi,\mathrm{Ad},\psi)$
explicitly, which fills in some blanks in the archimedean local theory of
Ginzburg's global integral.

Number Theory;Representation Theory