Let 
denote the number of primes of the form 
for 
, then
 |
(1)
|
where 
is the logarithmic integral (Shanks 1960,
pp. 321-332). Let 
denote the number of primes of
the form 
for 
, then
 |
(2)
|
(Shanks 1961, 1962). Let 
denote the number of pairs of primes 
and 
for 
, then
 |
(3)
|
where
 |
(4)
|
(Shanks 1960, pp. 201-203). Finally, let 
denote the number of pairs of primes 
and 
for 
, then
 |
(5)
|
(Lal 1967), where 
is called Lal's constant. Shanks (1967) showed that 
.
." Math. Comput. 21, 245-247, 1967.Shanks,
D. "On the Conjecture of Hardy and Littlewood Concerning the Number of Primes
of the Form 
."
Math. Comput. 14, 321-332, 1960.Shanks, D. "On Numbers
of the Form 
."
Math. Comput. 15, 186-189, 1961.Shanks, D. Corrigendum
to "On the Conjecture of Hardy and Littlewood Concerning the Number of Primes
of the Form 
."
Math. Comput. 16, 513, 1962.Shanks, D. "Lal's Constant
and Generalization." Math. Comput. 21, 705-707, 1967.