Serial correlation will not affect the value of the coefficients, instead what it influences is the standard error namely the t-value or the significance of the coefficient.

To test it, the most simple and famous statistic is DW value. The value lies in [0,4], if it is around 2 this indicates no serial correlation. While if it is near 0 or 4, it means there are either positive or negative serial correlation.

The drawback of DW test is it can only detect first order serial correlation. Some other method such as the lagrange multiplier can also test this.

best,