  We study D-branes wrapping an exceptional four-cycle P(1,a,b) in a blown-up
C^3/Z_m non-compact Calabi-Yau threefold with (m;a,b)=(3;1,1), (4;1,2) and
(6;2,3). In applying the method of local mirror symmetry we find that the
Picard-Fuchs equations for the local mirror periods in the Z_{3,4,6} orbifolds
take the same form as the ones in the local E_{6,7,8} del Pezzo models,
respectively. It is observed, however, that the orbifold models and the del
Pezzo models possess different physical properties because the background NS
B-field is turned on in the case of Z_{3,4,6} orbifolds. This is shown by
analyzing the periods and their monodromies in full detail with the help of
Meijer G-functions. We use the results to discuss D-brane configurations on
P(1,a,b) as well as on del Pezzo surfaces. We also discuss the number theoretic
aspect of local mirror symmetry and observe that the exponent which governs the
exponential growth of the Gromov-Witten invariants is determined by the special
value of the Dirichlet L-function.
