[math-fun] "Prime zeta function"

Is anything known about (let's dub it) "the prime zeta function" defined for Re(s) > 1 via Pzeta(s) := Sum{n=1..oo} of 1 / (p_n)^s, and elsewhere by analytic continuation, where as usual (p_n)^s := exp(s*ln(p_n)) ??? There's likewise the "alternating prime zeta function" defined for Re(s) > 1 via aPzeta(s) := Sum{n=1..oo} of (-1)^(n+1) / (p_n)^s I wonder what kind of a number aPzeta(1) = 1/2 - 1/3 + 1/5 - 1/7 + 1/11 - 1/13 + . . . = 0.26960+ is. --Dan