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We present a study of four muon, four electron, and two muon two electronevents using 1 fb^(-1) of data collected with the D0 detector at the FermilabTevatron p-barp Collider at sqrt(s) = 1.96 TeV. Requiring the lepton pairmasses to be greater than 30 GeV, we observe one event, consistent with theexpected background of 0.13 +- 0.03 events and with the predicted standardmodel ZZ and Z\gamma^* production of 1.71 +-0.15 events. We set an upper limiton the ZZ and Z\gamma^* cross section of 4.4 pb at the 95% C.L. We also derivelimits on anomalous neutral trilinear ZZZ and ZZ\gamma^* gauge couplings. Theone-parameter 95%$ C.L. coupling limits with a form factor scale Lambda = 1.2TeV are -0.28 < f_(40)^Z < 0.28, -0.31 < f_(50)^Z < 0.29, -0.26 < f_(40)^\gamma< 0.26, and -0.30 < f_(5 0)^\gamma < 0.28.

Search for ZZ and Z\gamma^* production in p-barp collisions at sqrt(s) = 1.96 TeV and limits on anomalous ZZZ and ZZ\gamma^* couplings