There is an old conjecture of Chebyshev saying that there are more primes $p\equiv 3 \pmod 4$ than $p\equiv 1 \pmod 4$. S. Knapowski and P. Turán have given a quantitative interpretation of this statement. We prove a theorem about the boudary values of general Dirichlet series and show its relevance to Chebyshev's problem. In particular it turns out that the Knapowski-Turán coujecture is false at least if we accept the Riemann Hypothesis for $L$-functions $\pmod 4$.