The line \(L_2\) passing through \((3, -1)\) divides the line segment \(L_1\) joining the points \((-1, 2)\) and \((3, 6)\) in the ratio \(1 : 3\) internally. The equation of line \(L_2\) is:

Let the given points be \(A(-1, 2)\) and \(B(3, 6)\).

Let \(P\) be the point that divides the line segment \(AB\) internally in the ratio \(1 : 3\).

Using the section formula, the coordinates of \(P\) are given by:

\(x = \dfrac{1(3) + 3(-1)}{1 + 3} = \dfrac{3 - 3}{4} = 0\)

\(y = \dfrac{1(6) + 3(2)}{1 + 3} = \dfrac{6 + 6}{4} = 3\)

Thus, the coordinates of \(P\) are \((0, 3)\).

The line \(L_2\) passes through the points \((3, -1)\) and \((0, 3)\).

The slope of the line \(L_2\) is:

\(m = \dfrac{3 - (-1)}{0 - 3} = -\dfrac{4}{3}\)

The equation of the line \(L_2\) is:

\(y - 3 = -\dfrac{4}{3}(x - 0)\)

\(3y - 9 = -4x\)

\(4x + 3y - 9 = 0\)

Answer: \(4x + 3y - 9 = 0\)