Is there a set of triple-primes satisfying the following equation?

$p_1p_2+p_2p_3+p_3p_1+p_1+p_2+p_3=2^β,p_1p_2p_3=2^α-1,α>β.$ I have checked the first 11 numbers that no one satisfy the above condition. It's easy to check that when all the prime numbers $p_i$ are in the form of $4k+3,$ then we have $p_i=2^{s_{i}}k_i+2^{s_{i}}−1$