Lorentz transformations when the velocity is not along the x-axis.

Does anyone know the equations for the Lorentz transformations when the
velocity is not along the x-axis only. I have been searching the
internet, but with no success, so I have attempted to derive these
equations myself, and have came up with the following:

t' = c.(t - v.sqrt(x^2 + y^2 + z^2) / c^2) / sqrt(c^2 - v^2)
x' = c.(x - x.v.t / sqrt(x^2 + y^2 + z^2)) / sqrt(c^2 - v^2)
y' = c.(y - y.v.t / sqrt(x^2 + y^2 + z^2)) / sqrt(c^2 - v^2)
z' = c.(z - z.v.t / sqrt(x^2 + y^2 + z^2)) / sqrt(c^2 - v^2)

where, gamma = c / sqrt(c^2 - v^2)

Can someone tell me if these equations are valid?

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