The present note reports on an explicit spectral formula for
the fourth moment of the Dedekind zeta function $\zeta_{\mathrm{F}}$
of the Gaussian number field $\mathrm{F} = \mathbf{Q}(i)$,
and on a new version of the sum formula of Kuznetsov type for
$\mathrm{PSL}_2(\mathbf{Z}[i])\backslash \mathrm{PSL}_2(\mathbf{C})$.
Our explicit formula (Theorem 5, below) for $\zeta_{\mathrm{F}}$
gives rise to a solution to a problem that has been posed on
p. 183 of [M3] and, more explicitly, in [M4]. Also, our sum
formula (Theorem 4, below) is an answer to a problem raised
in [M4] concerning the inversion of a spectral sum formula
over the Picard group $\mathrm{PSL}_2(\mathbf{Z}[i])$ acting
on the three dimensional hyperbolic space (the $K$-trivial
situation). To solve this problem, it was necessary to include
the $K$-nontrivial situation into consideration, which is analogous
to what has been experienced in the modular case.