Doping dependence of the spatially modulated dynamical spin correlations and the superconducting-transition temperature in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">La</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi mathvariant="normal">−</mml:mi><mml:mi mathvariant="italic">x</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">Sr</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">CuO</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

Systematic low-energy neutron-scattering studies have been performed on float-zone-grown single crystals of ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ with $x$ extending from zero doping, $x=0$, to the overdoped, weakly superconducting regime, $x=0.25$. For $x$ beyond a critical doping value of ${x}_{c}\ensuremath{\approx}0.05$ the low-energy spin-fluctuation peak position shifts from ($\frac{1}{2}$, $\frac{1}{2}$) to ($\frac{1}{2}$\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}, $\frac{1}{2}$), and ($\frac{1}{2}$, $\frac{1}{2}$\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}); ${x}_{c}$ also represents the onset concentration for superconductivity. For $0.06&lt;~x&lt;~0.12$ the incommensurability $\ensuremath{\delta}$ follows approximately the quantitative relation $\ensuremath{\delta}=x$. However, beyond $x\ensuremath{\approx}0.12$ the incommensurability tends to saturate around $\ensuremath{\delta}\ensuremath{\approx}1/8$. The superconducting-transition temperature ${T}_{c}(x)$ for stoichiometric samples at a given doping scales linearly with $\ensuremath{\delta}$ up to the optimal doping value of $x$. The peak momentum width of the spin fluctuations at low energies is small throughout the superconducting concentration region except in the strongly overdoped region. An anomalously small width is observed for $x=\frac{1}{8}$. The incommensurate spatial modulation is found to be robust with respect to pair-breaking effects that lower ${T}_{c},$ such as deoxygenation of the sample or replacement of Cu by Zn.