Calculate the difference between two error functions with argument having different real parts and same imaginary part

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Does anyone know how to calculate the difference between two error functions with arguments having different real parts and the same imaginary part?

Basically I'm looking to simplify:

$$\mathrm{erf}(L_2 + \mathrm{i} a) - \mathrm{erf}(L_1+\mathrm{i}a).$$