Let $q_n$ be the $n$-th number in
$\{\,p\in\mathbb P\,|\,p\equiv 3\pmod 4\,\}$ and define
$r_n=\frac{\pi^2}{4}n\ln n$, then
$1\leq n\leq 1973\implies r_n\leq q_n$.
$\qquad\quad q_n$ is the blue curve and $r_n$ is the red curve.
But $r_{1974}=36958>36947=q_{1974},\;r_{10000}-q_{10000}=2989$ and $r_{100000}-q_{100000}=93029$. For $n>1800$ prime numbers of this form starts to come closer and the nice formula for $r_n$ loose all its accuracy.
n qn rn
1 3 0
2 7 3
3 11 8
4 19 14
5 23 20
6 31 27
7 43 34
8 47 41
9 59 49
10 67 57
11 71 65
12 79 74
13 83 82
14 103 91
15 107 100
16 127 109
17 131 119
18 139 128
19 151 138
20 163 148
Are there known formulas to estimate $q_n$?
Guided by a comment I managed to look up the formula $\pi(x;d,a)\sim\frac{x}{\varphi(d)\ln x}$, where $\varphi(d)$ is the same for primes of the form $nd+a$. In this case $\varphi(4)=2$ is the constant to replace $\frac{\pi^2}{4}$.
The formula above comes from A Computational Introduction to Number Theory and Algebra, Victor Shoup
