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As far as I know, the April 2011 version of #143 on this page has not been improved upon.
On page 10 of that paper, the authors give an algorithm that uses a constant $\:c_{\hspace{.01 in}5}\:$.
According to page 2 and page 6 of that paper, $\:c_{\hspace{.01 in}5}\:$ should be effectively computable.

Has anyone actually computed a value of $\:c_{\hspace{.01 in}5}\:$ that will work?

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Your question is about the result of Deshouillers–Iwaniec in the paper "On the Brun-Titchmarsh theorem on average", $$ \pi(x,q,a)\le \frac{(\frac{4}{3}+\epsilon c_5)x}{\phi(x)\log (\frac{x}{q})}, $$ where the notations are explained in the above paper of Carl Pomerance. It is indeed mentioned that the constants are (in principle) effectively computable, but on the other hand this appears to be difficult (with no appaerent progress from $2005$ version to the $2011$ version). I am no expert, but it appears to me that there has been no value computed here explicitly. In fact, Harman writes in http://www.ams.org/journals/mcom/2005-74-252/S0025-5718-05-01749-7/S0025-5718-05-01749-7.pdf the following (see Lemma $1$): "Better bounds than the factor $2$ (Brun-Titchmarsh) can be proved for the parameters in certain ranges using deep results on averages of Kloosterman sums, and this leads to problems when trying to calculate the constants involved".
Hence my impression is, that this has not yet been achieved.

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