  We present a sample of 373 peaked-spectrum (PS) sources with spectral peaks
around 150MHz, selected using a subset of two LOFAR all-sky surveys, the LOFAR
Two Meter Sky Survey and the LOFAR LBA Sky Survey. These surveys are the most
sensitive low-frequency widefield surveys to date, allowing us to select
low-luminosity PS sources. Our sample increases the number of known PS sources
in our survey area by a factor 50. The 5GHz luminosity distribution of our PS
sample shows we sample the lowest luminosity PS sources to-date by nearly an
order of magnitude. Since high-frequency PS sources and compact steep-spectrum
sources are hypothesised to be the precursors to large radio galaxies, we
investigate whether this is also the case for our sample of low-frequency PS
sources. Using optical line emission criteria, we find that our PS sources are
predominately high-excitation radio galaxies instead of low-excitation radio
galaxies, corresponding to a quickly evolving population. We compute the radio
source counts of our PS sample, and find they are scaled down by a factor of
$\sim$40 compared to a general sample of radio-loud active galactic nuclei
(AGN). This implies that the lifetimes of PS sources are 40 times shorter than
large scale radio galaxies, if their luminosity functions are identical. To
investigate this, we compute the first radio luminosity function for a
homogeneously-selected PS sample. We find that for 144MHz luminosities $\gtrsim
10^{25}$W Hz$^{-1}$, the PS luminosity function has the same shape as an
unresolved radio-loud AGN population but shifted down by a factor of $\sim$10.
We interpret this as strong evidence that these high-luminosity PS sources
evolve into large-scale radio-loud AGN. For local, low-luminosity PS sources,
there is a surplus of PS sources, which we hypothesise to be the addition of
frustrated PS sources that do not evolve into large-scale AGN.
