  By applying the formula for essential Whittaker functions established by
Matringe and Miyauchi, we study five integral representations for irreducible
admissible generic representations of ${\rm GL}_n$ over $p$-adic fields. In
each case, we show that the integrals achieve local formal $L$-functions
defined by Langlands parameters, when the test vector is associated to the
newform. We give the relation between local periods involving essential
Whittaker functions and special values of formal $L$-factors at $s=1$ for
certain distinguished or unitary representations. The period integrals are also
served as standard non-zero distinguished forms.
