the initial conditions are at the same time --you want L(t0) and dL/dt(t0)
If you think about it suppose you were doing projectile motion then
you would need the position s(t) and velocity v(t) both at t = 0
For L = k1t^1/3 + k2t^2/3
L(t0) = L0 = k1(t0)^1/3 + k2(t0)^2/3
dL/dt(t0) = 1/3k1(t0)^(-2/3) +2/3k2(t0)^-1/3
do you have any other info on dL/dt(t0) ?