Checkout Theorem 5.1 of this Bickel et al.. A statistically optimal choice in terms of the erroerror $\|y-\hat{y}\|_2^2$$\|y-\hat{y}(\lambda)\|_2^2$ is $\lambda = A \sigma_{\text{noise}} \left(\frac{\log p}{n}\right)^{/12}$$\lambda = A \sigma_{\text{noise}} \sqrt{\dfrac{\log p}{n}}$ (with high probability), for a constant $A > 2\sqrt{2}$.