How to solve sparse hermitian banded Matrix?

I’m looking for a way to solve a slight extension of what gtsv can offer - I want to solve Ax=b where A is a (1e61e6) symmetric pos def banded matrix, with 2 sub+super diagonals (instead of 1), and b is (1e620) i.e 20 right hand sides

Assuming I cannot factorise the solve into a set of tridiagaonal solves, what would be the fastest way to accomplish this, and is it do-able within even an order of magnitude of gtsv speed?