By far the largest experiment (as far as I know) conducted on a double blind basis to prove the power of prayer (or otherwise) was $2.4 million spent by the Templeton Foundation. 'Study of the therapeutic effects of intercessory prayer (STEP) in cardiac bypass patients'.American Heart Journal 151:4, 2006 934 - 942

There were three groups: Group 1 received prayers and didn't know it; Group 2 received no prayers and didn't know it; Group three received prayers and did know it.

Prayers were delivered by three congregations distant from the hospitals.

The results were clear cut. There was no difference between those prayed for and those not. There was a difference between those who knew they were being prayed for and those who did not know one way of the other, but it went in the wrong direction. Those who knew they had been the beneficiaries of prayer suffered significantly more complications than those who did not.

The experimenters suggested they were stressed out because of "performance anxiety".