I've been experimenting with some infinite series, and I've been looking at this one, $\sum_{k=1}^\infty (-1)^{k+1} {1\over p_k}$ where $p_k$ is the k-th prime. I've summed up the first 35 terms myself and got a value of about 0.27935, and this doesn't seem close to a relation of any 'special' constants, except maybe $\frac12\gamma $.
My question is, has the sum of this series been proven to have a particular closed form? If so, what is this value?