The families of motives of the title arise from classical one-variable hypergeometric functions. This talk will focus on the calculation of their corresponding L-functions both in theory and in practice. These L-functions provide a fairly wide class which is numerically accessible. As an illustration we will consider the case of Artin L-functions. In the main example the corresponding Galois group is a subgroup of the Weyl group of F_4 of order 1152. The associated degree four L-functions are related to the lines in certain affine cubic surfaces. This is joint work with H. Cohen.