Complex Unsymmetric Banded Matrix Diagonalization

[lubasch]

Hallo,
does LAPACK have a subroutine that solves the eigenvalue problem for a banded complex unsymmetric (=general) matrix?
In my matrix only the diagonal and 4 off-diagonals are non-zero, all entries are complex and the matrix is unsymmetric.
ZGEEV and ZGEEVX are too slow (even Mathematica is faster). If there are no other routines, where can I find code that does this job?
Thanks a lot in advance for any response!
Michael