The 7 nonthermal filamentary structures located in the Galactic center region (l < \1 degrees\) are believed to manifest strong (few mG) vertical magnetic field lines coherent over the tens of parsec scale. We investigate the nature of this large-scale component with measurements of Zeeman splitting, in the 1665 and 1667 MHz absorption lines of OH, toward 13 selected positions: Clump 2, Sgr D, Sgr B2 and B1, the ''l = 1 degrees.5 complex'', the linear and arched filaments of the Galactic center radio are (GCRA), 1E1740.7-2942 (The Great Annihilator), the nonthermal filaments G359.54+0.18 and G359.1-0.2, and portions of the expanding molecular ring (or the innermost X(1) bar orbit). The results are surprising indeed - we detect no line-of-sight large scale magnetic field, above an average 3 sigma upper limit of similar to 0.3 mG, toward the observed positions. We conclude that the strong vertical magnetic field lines are contained in localized bundles which are, (1) aligned nearly within the plane of the sky (+/-5 degrees), thus showing no line-of-sight B component and/or, (2) are not strongly coupled to the relatively dense (10(3) cm(-3)) molecular gas sampled by this study.