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Let      θ  ( x ; 3 , l ) \theta (x;3,l)       be the sum of the logarithms of the primes not exceeding  x  that are congruent to  l  modulo 3, where  l  is 1 or 2. By the prime number theorem for arithmetic progressions,      θ  ( x ; 3 , l ) ∼  x / 2 \theta (x;3,l) \sim x/2       as      x →  ∞  x \to \infty      . Using information concerning zeros of Dirichlet  L -functions, we prove explicit numerical bounds for      θ  ( x ; 3 , l ) \theta (x;3,l)       of the form      | θ  ( x ; 3 , l ) −  x / 2 | &gt; e x |\theta (x;3,l) - x/2| &gt; ex      ,      x ⩾  x 0 ( ε  ) x \geqslant {x_0}(\varepsilon )      .

Explicit estimates for 𝜃(𝑥;3,𝑙) and 𝜓(𝑥;3,𝑙)