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This is an English translation of the Latin original "De summa seriei exnumeris primis formatae${1/3}-{1/5}+{1/7}+{1/11}-{1/13}-{1/17}+{1/19}+{1/23}-{1/29}+{1/31}-$ etc. ubinumeri primi formae $4n-1$ habent signum positivum formae autem $4n+1$ signumnegativum" (1775). E596 in the Enestrom index.  Let $\chi$ be the nontrivial character modulo 4. Euler wants to know what$\sum_p \chi(p)/p$ is, either an exact expression or an approximation. He looksfor analogies to the harmonic series and the series of reciprocals of theprimes.  Another reason he is interested in this is that if this series has a finitevalue (which is does, the best approximation Euler gets is 0.3349816 in section27) then there are infinitely many primes congruent to 1 mod 4 and infinitelymany primes congruent to 3 mod 4.  In section 15 Euler gives the Euler product for the L(chi,1).  As a modern mathematical appendix appendix, I have written a proof followingDavenport that the series $\sum_p \frac{\chi(p)}{p}$ converges. This involvesapplications of summation by parts, and uses Chebyshev's estimate for thesecond Chebyshev function (summing the von Mangoldt function).

On the sum of the series formed from the prime numbers where the prime  numbers of the form $4n-1$ have a positive sign and those of the form $4n+1$  a negative sign