Python, 5333 50454991
I believe this is the first contender to score significantly better than a random oracle.
def H(s):n=int(s.encode('hex'),16);return n%(8**8-ord('Z%`5K'+%:5O!([/5;O5hXsI,]\'k[5;QwrXsIf]\'k#!}_u5O__u5O}zuSC;nQ~{;/~FCu8%;7cu"A~{uUkwO"u=-.Me*CnCutM;ItulA{u`bluPb'[n%70]uOk_7"ud-o?y<Cn~-`bl_Yb'[n%70]))