Can You do the Shmoneh Esrai without washing?

One must wash one’s hands-without a Bracha- before davening. If, however, someone has said Shmone Esrai without washing, he should not repeat the Shmone Esrai. Someone who has washed his hands upon awakening must wash them again before davening only if a considerable time has elapsed and absent-mindedly he may have touched parts of his body that are normally covered (Shulchan Aruch Orach Chayim 92,4-5).