  The Hirzebruch functional equation is \[
  \sum_{i = 1}^{n} \prod_{j \ne i} { 1 \over f(z_j - z_i)} = c \] with constant
$c$ and initial conditions $f(0)=0, f'(0)=1$. In this paper we find all
solutions of the Hirzebruch functional equation for $n \leqslant 6$ in the
class of meromorphic functions and in the class of series. Previously, such
results were known only for $n \leqslant 4$.
  The Todd function is the function determining the two-parametric Todd genus
(i.e. the $\chi_{a,b}$-genus). It gives a solution to the Hirzebruch functional
equation for any $n$. The elliptic function of level $N$ is the function
determining the elliptic genus of level $N$. It gives a solution to the
Hirzebruch functional equation for $n$ divisible by $N$.
  A series corresponding to a meromorphic function $f$ with parameters in $U
\subset \mathbb{C}^k$ is a series with parameters in the Zariski closure of $U$
in $\mathbb{C}^k$, such that for parameters in $U$ it coincides with the series
expansion at zero of $f$. The main results are:
  Any series solution of the Hirzebruch functional equation for $n = 5$
corresponds to the Todd function or to the elliptic function of level $5$.
  Any series solution of the Hirzebruch functional equation for $n = 6$
corresponds to the Todd function or to the elliptic function of level $2$, $3$
or $6$.
  This gives a complete classification of complex genera that are fiber
multiplicative with respect to $\mathbb{C}P^{n-1}$ for $n \leqslant 6$.
