In his Ph.D thesis, John Tate attached local root numbers with characters of a non-Archimedean local field of characteristic zero. Robert Langlands (later P. Deligne ) proved the existence theorem of non-abelian local root numbers of higher dimensional complex local Galois representations. The local Langlands correspondence preserves thisroot numbers and the global root number is a product of local root numbers. So the explicit computation of the local root numbersis an integral part of the Langlands programs. But for arbitrary higher dimensional Galois representations we do not have any explicit formula for the local root numbers. To give an explicit formula of the local root number of an induced representation of a local Galois group of a non-Archimedean local field F ofcharacteristic zero, first we have to compute the Langlands' lambda function \lambda_{K/F} for a finite extension K/F . The plan of the talk is as follows:

1. Review of local and global root numbers
2. To explain the computations of the local root numbers for a particular local Galois representations (e.g. Heisenberg representations)
3. And the computation of the lambda function \lambda_{K/F} except when K/F is quadratic wildly ramified extensions
4. If time permits, then I will explain some of my ongoing projects regarding root numbers and some open problems.