  For any discretely self-similar, incompressible initial data $v_0$ which
satisfies $\|v_0 \|_{L^3_w(\mathbb R^3)}\leq c_0$ where $c_0$ is allowed to be
large, we construct a forward discretely self-similar local Leray solution in
the sense of Lemari\'e-Rieusset to the 3D Navier-Stokes equations in the whole
space. No further assumptions are imposed on the initial data; in particular,
the data is not required to be continuous or locally bounded on $\mathbb
R^3\setminus \{0\}$. The same method is used to construct self-similar
solutions for any $-1$-homogeneous initial data in $L^3_w$.
