The small-scale dynamo is a process by which turbulent kinetic energy is converted into magnetic energy, and thus it is expected to depend crucially on the nature of the turbulence. In this paper, we present a model for the small-scale dynamo that takes into account the slope of the turbulent velocity spectrum $v\left(\ell \right)\propto {\ell }^{\vartheta }$, where $\ell$ and $v\left(\ell \right)$ are the size of a turbulent fluctuation and the typical velocity on that scale. The time evolution of the fluctuation component of the magnetic field, i.e., the small-scale field, is described by the Kazantsev equation. We solve this linear differential equation for its eigenvalues with the quantum-mechanical WKB approximation. The validity of this method is estimated as a function of the magnetic Prandtl number Pm. We calculate the minimal magnetic Reynolds number for dynamo action, ${\mathrm{Rm}}_{\mathrm{crit}}$, using our model of the turbulent velocity correlation function. For Kolmogorov turbulence ($\vartheta =1/3$), we find that the critical magnetic Reynolds number is ${\mathrm{Rm}}_{\mathrm{crit}}^{\mathrm{K}}\approx 110$ and for Burgers turbulence ($\vartheta =1/2$) ${\mathrm{Rm}}_{\mathrm{crit}}^{\mathrm{B}}\approx 2700$. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is $\Gamma \propto {\mathrm{Re}}^{(1-\vartheta )/(1+\vartheta )}$ in the limit of infinite magnetic Prandtl number. For decreasing magnetic Prandtl number (down to $\mathrm{Pm}\gtrsim 10$), the growth rate of the small-scale dynamo decreases. The details of this drop depend on the WKB approximation, which becomes invalid for a magnetic Prandtl number of about unity.

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