It is well known that the sum of all inverse primes is divergent. But the alternating sum is convergent by the Leiniz criterion. To which known constant "a" does the sum converge?
$$a = \frac{1}{2} - \frac{1}{3} +\frac{1}{5}-\frac{1}{7}+\frac{1}{11} -+ ...$$
For clarity and completeness, I have put the information in the comments into an answer...
As mentioned, this series has an expansion given by the OEIS, which states that the most accurate known estimate of the limit is 0.26960635197167...
The references given therein, as well as others such as Mathworld, Wells, Robinson & Potter and Weisstein indicate that no known closed form of this limit is known, nor does it have its own special name or symbol.