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In this paper, the first part of a larger work, we prove the spectral decomposition of $$ \int_{-\infty}^\infty|\zeta(\s+it)|^4g(t){\rm d}t\qquad(\hf < \sigma < 1 {\rm {fixed}}), $$ where $g(t)$ is a suitable weight function of fast decay. This is used to obtain estimates and omega results for the function $$\eqalign{E_2(T,\sigma) &: =\int_0^T|\zeta(\sigma+it)|^4{rm d}t - {\zeta^4(2\sigma)\over\zeta(4\sigma)}T -{T\over3-4\sigma}{({T\over2\pi} )}^{2-4\sigma}{\zeta^4(2-2\sigma)\over\zeta(4-4\sigma)}\cr& - T^{2-2\sigma}(a_0(\sigma) + a_1(\sigma)\log T + a_2(\sigma)\log^2T),\cr} $$ the error term in the asymptotic formula for the fourth moment of $|\zeta(\sigma+it)|$.

The Moments of the Riemann Zeta-Function Part I: The fourth moment off the critical line