  We derive precise formulas for the archimedean Euler factors occurring in
certain standard Langlands $L$-functions for unitary groups. In the 1980s, Paul
Garrett, as well as Ilya Piatetski-Shapiro and Stephen Rallis (independently of
Garrett), discovered integral representations of automorphic $L$-functions that
are Eulerian but, in contrast to the Rankin--Selberg and Langlands--Shahidi
methods, do not require that the automorphic representations to which the
$L$-functions are associated are globally generic. Their approach, the doubling
method, opened the door to a variety of applications that could not be handled
by prior methods.
  For over three decades, though, the integrals occurring in the Euler factors
at archimedean places for unitary groups eluded precise computation, except
under particular simplifications (such as requiring certain representations to
be one-dimensional, as Garrett did in the first major progress on this
computation and only prior progress for general signatures). We compute these
integrals for holomorphic discrete series of general vector weights for unitary
groups of any signature. This has consequences not only for special values of
$L$-functions in the archimedean setting, but also for $p$-adic $L$-functions,
where the corresponding term had remained open.
