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In metric of spaces $L_{s}, \ 1< s\leq\infty$, we obtain exact orderestimates of best approximations and approximations by Fourier sums of classesof convolutions the periodic functions that belong to unit ball of space$L_{1}$, with generating kernel$\Psi_{\beta}(t)=\sum\limits_{k=1}^{\infty}\psi(k)\cos(kt-\frac{\beta\pi}{2})$,$\beta\in\mathbb{R}$, whose coefficients $\psi(k)$ are such that product$\psi(n)n^{1-\frac{1}{s}}$, $1<s\leq\infty$, can't tend to nought faster thanevery power function and besides, if $1<s<\infty$, then$\sum\limits_{k=1}^{\infty}\psi^{s}(k)k^{s-2}<\infty$ and if $s=\infty$, then$\sum\limits_{k=1}^{\infty}\psi(k)<\infty$.

Estimates of the best approximations and approximations of Fourier sums  of classes of convolutions of periodic functions of not high smoothness in  integral metrics