Letf（x）∈C[-1,1],pn*（x）bethebestapproximationpolynomialofdegreentof（x）.G.Iorentzconjecturedthatifforalln,p2n*（x）=p2n+1*（x）,thenfiseven;andifp2n+1*（x）=p2n+2*（x）,po*（z）=0,thenfisodd.Inthispaper,itisprovedthat,undertheL1-norm,theLorentzconjectureisvalidconditionally,i.e.if（i）（1-x2）f（x）canbeextendedtoanabsolutelyconvergentTehebyshevsories;（ii）foreveryn,f（x）-p2n+1*（x）hasexactly2n+2zeros(or,inthearcondsituation,f（x）-p2n+2*（x）hasexaetly2n+3zeros),thenLorentzconjectureisvalid.
