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For the one-dimensional long-ranged Ising spin-glass with random couplingsdecaying with the distance $r$ as $J(r) \sim r^{-\sigma}$ and distributed withthe L\'evy symmetric stable distribution of index $1 <\mu \leq 2$ (includingthe usual Gaussian case $\mu=2$), we consider the region $\sigma>1/\mu$ wherethe energy is extensive. We study two real space renormalization procedures atzero temperature, namely a simple box decimation that leads to explicitcalculations, and a strong disorder decimation that can be studied numericallyon large sizes. The droplet exponent governing the scaling of the renormalizedcouplings $J_L \propto L^{\theta_{\mu}(\sigma)}$ is found to be$\theta_{\mu}(\sigma)=\frac{2}{\mu}-\sigma$ whenever the long-ranged couplingsare relevant $\theta_{\mu}(\sigma)=\frac{2}{\mu}-\sigma \geq -1$. For thestatistics of the ground state energy $E_L^{GS}$ over disordered samples, weobtain that the droplet exponent $\theta_{\mu}(\sigma) $ governs the leadingcorrection to extensivity of the averaged value $\overline{E_L^{GS}} \simeq Le_0 +L^{\theta_{\mu}(\sigma)} e_1$. The characteristic scale of thefluctuations around this average is of order $L^{\frac{1}{\mu}}$, and therescaled variable $u=(E_L^{GS}-\overline{E_L^{GS}})/L^{\frac{1}{\mu}}$ isGaussian distributed for $\mu=2$, or displays the negative power-law tail in$1/(-u)^{1+\mu}$ for $u \to -\infty$ in the L\'evy case $1<\mu<2$.

One-dimensional Ising spin-glass with power-law interaction : real-space renormalization at zero temperature