  In analytic number theory, several results make use of information regarding
the prime values of a multiplicative function in order to extract information
about its averages. Examples of such results include Wirsing's theorem and the
Landau-Selberg-Delange method. In this paper, we are interested in the opposite
direction. In particular, we prove that when $f$ is a suitable divisor-bounded
multiplicative function with small partial sums, then
$f(p)\approx-p^{i\gamma_1}-\ldots-p^{i\gamma_m}$ on average, where the
$\gamma_j$'s are the imaginary parts of the zeros of the Dirichet series of $f$
on the line $\Re(s)=1$. This extends a result of Koukoulopoulos and
Soundararajan and it builds upon ideas coming from previous work of
Koukoulopoulos for the case where $|f|\leqslant 1$.
