What do you mean by L, B, and R?
The meaning of the symbols in the geodesic equations are this:
\alpha, \beta, \gamma are just placeholders, that take on the values (0,1,2,3)So x^{\alpha} stands for the set (x^0, x^1, x^2, x^3)
This is a vector, it's got four elements though, not the three you might be used to. One of them is time.
It's got four components, one of them is time, and three are space. The exact correspondence can vary, but usually it will be
x^0 -> t x^1->r x^2->\theta x^3->\phi
The geodesics are parameterized in terms of an affine parameter, usually either proper time \tau, or more generally \lambda.
For now, you can think of \lambda as just a number. Each value of \lambda specifies one point on the geodesic curve, sweeping it through it's allowed range generates the complete curve.
We don't write the position as a function of time. Instead, we write the position and the time as a function of the parameter lambda.So, going back to the original equation
<br />
\frac{d^{2}x^{\alpha }}{d\lambda ^{2}} + \Gamma ^{\alpha }_{\beta \gamma }\frac{\mathrm{d} x^{\beta }}{\mathrm{d} \lambda }\frac{\mathrm{d} x^{\gamma }}{\mathrm{d} \lambda } = 0<br />
This is a compact way of writing four equations, one equation for each value of \alpha
The repeated indices, \beta and \gamma imply a summation.
So it should really be written as
<br />
\sum_{\beta=0}^{4} \sum_{\gamma=0}^{4}\frac{d^{2}x^{\alpha }}{d\lambda ^{2}} + \Gamma ^{\alpha }_{\beta \gamma }\frac{\mathrm{d} x^{\beta }}{\mathrm{d} \lambda }\frac{\mathrm{d} x^{\gamma }}{\mathrm{d} \lambda } = 0<br />
but that's too much work, so people use the "Einstein summation convention" and omit the sums for repeated indices.
So this is really a set of four differential equations.
