I went through the physics program at the university of Minnesota and the point at which the rigorous nature of the scientific process really hit home for me was when I successfully reproduced the results of the famous Milikan Oil Drop experiment for measuring the fundamental charge e and having my professor tell me that even that experiment had controversy over whether Milikin was "picking bubbles" with bias. Real science is hardcore; you better have phenomenal data and explain every potential source of errors and what exatly those error margins are.
tl;dr - There is an author/___domain expert in Intercessory Prayer claiming there is scientific evidence backing up the benefit of praying or thinking positively on behalf of something/someone. This article proves the evidence is flawed based on scientific process and concludes the evidence that isn't flawed points to a non-correlation between intercessory prayer and how it can help. He does say that religion and science don't have to mix.
Personally I don't really like the wording of 'extraordinary evidence'. In fact, for most extraordinary claims the level of evidence the scientific community requires isn't that extraordinary at all.
There is also a very real danger of misinterpreting the adjective 'extraordinary'. Take for example statistics. It might be tempting to say that a higher correlation and/or significance is more extraordinary, but I would say any significant result should suffice.
My point is that, 'extraordinary' should not apply to the result, but to the experiment itself. To test an (extraordinary) hypothesis, we should design experiments as if we really want to disprove the hypothesis.
Background:
Telomerase activity is a predictor of long-term cellular viability, which decreases with chronic psychological distress (Epel et al., 2004). Buddhist traditions claim that meditation decreases psychological distress and promotes well-being (e.g., Dalai Lama and Cutler, 2009). Therefore, we investigated the effects of a 3-month meditation retreat on telomerase activity and two major contributors to the experience of stress: Perceived Control (associated with decreased stress) and Neuroticism (associated with increased subjective distress). We used mediation models to test whether changes in Perceived Control and Neuroticism explained meditation retreat effects on telomerase activity. In addition, we investigated whether two qualities developed by meditative practice, increased Mindfulness and Purpose in Life, accounted for retreat-related changes in the two stress-related variables and in telomerase activity.
When someone makes an extraordinary clain I require an extraorderinary level evidence. I think that praying to a supernatural entity for help qualifies as an extraordinary claim. So far the evidence has not matched the level of the claim.