Le costanti di Eulero – Lehmer γ(a, q) sono una generalizzazione della costante di Eulero γ. Sono definite come 
, per 0 ≤ a < q e q > 1.
Alcune formule che coinvolgono le costanti di Eulero – Lehmer:
;
;
;
;
se MCD(a, q) = d, 
.
M. Ram Murty e N. Saradha dimostrarono nel 2010 che le costanti γ(a, q) per 0 < a < q e q > 1 sono tutte trascendenti, tranne al massimo una. Dato che 
, se γ è algebrica (cosa che nessun esperto crede possibile), allora l’unica costante di Eulero – Lehmer algebrica è γ(2, 4).
Ram Murty e Saradha dimostrarono anche che:
-
se a, q1, q2, … qn è una sequenza strettamente crescente di interi positivi tutti primi tra loro, vi sono al massimo due numeri uguali tra γ e tutti i vari γ(a, qk);
-
vi sono al massimo due numeri uguali tra γ e tutti i vari
per 0 < a < q e q > 1, escludendo γ(2, 4).
Le tabelle seguenti mostrano i valori di γ(a, q) per 0 ≤ a < q e q fino a 20.
|
q \ a |
0 |
1 |
2 |
3 |
4 |
5 |
|
1 |
0.5772156649 |
- |
- |
- |
- |
- |
|
2 |
–0.0579657578 |
0.6351814227 |
- |
- |
- |
- |
|
3 |
–0.1737988746 |
0.6778071638 |
0.0732073757 |
- |
- |
- |
|
4 |
–0.2022696741 |
0.7102897931 |
0.1443039162 |
–0.0751083703 |
- |
- |
|
5 |
–0.2064444495 |
0.7359203968 |
0.1903893264 |
–0.0137637397 |
–0.1288858691 |
- |
|
6 |
–0.2024239674 |
0.7567280060 |
0.2233790518 |
0.0286250928 |
–0.0789208422 |
–0.1501716761 |
|
7 |
–0.1955277835 |
0.7740100133 |
0.2485154045 |
0.0599869440 |
–0.0424484877 |
–0.1093898155 |
|
8 |
–0.1877782346 |
0.7886313902 |
0.2685014990 |
0.0843196884 |
–0.0144914395 |
–0.0783415971 |
|
9 |
–0.1800009903 |
0.8011909666 |
0.2848900550 |
0.1038676892 |
0.0077266234 |
–0.0538231041 |
|
10 |
–0.1725369428 |
0.8121169847 |
0.2986454804 |
0.1199939129 |
0.0258799452 |
–0.0339075067 |
|
11 |
–0.1655163280 |
0.8217242863 |
0.3104032225 |
0.1335773143 |
0.0410403532 |
–0.0173650319 |
|
12 |
–0.1589742487 |
0.8302498899 |
0.3206017380 |
0.1452122403 |
0.0539272609 |
–0.0033729493 |
|
13 |
–0.1529025917 |
0.8378759079 |
0.3295549454 |
0.1553163023 |
0.0650424447 |
0.0086406213 |
|
14 |
–0.1472744046 |
0.8447447701 |
0.3374944938 |
0.1641923647 |
0.0747471893 |
0.0190860025 |
|
15 |
–0.1420556357 |
0.8509696137 |
0.3445956778 |
0.1720659797 |
0.0833086756 |
0.0282655719 |
|
16 |
–0.1372108161 |
0.8566415480 |
0.3509939963 |
0.1791089538 |
0.0909290507 |
0.0364073034 |
|
17 |
–0.1327057458 |
0.8618348412 |
0.3567961092 |
0.1854547166 |
0.0977642960 |
0.0436864965 |
|
18 |
–0.1285086718 |
0.8666106954 |
0.3620873066 |
0.1912086526 |
0.1039368508 |
0.0502402857 |
|
19 |
–0.1245907008 |
0.8710200409 |
0.3669367344 |
0.1964552331 |
0.1095442824 |
0.0561775849 |
|
20 |
–0.1209258304 |
0.8751056359 |
0.3714011333 |
0.2012630536 |
0.1146653812 |
0.0615860630 |
|
q \ a |
6 |
7 |
8 |
9 |
10 |
11 |
|
7 |
–0.1579306102 |
- |
- |
- |
- |
- |
|
8 |
–0.1241975827 |
–0.1594280588 |
- |
- |
- |
- |
|
9 |
–0.0976655735 |
–0.1311104262 |
–0.1578595752 |
- |
- |
- |
|
10 |
–0.0761965879 |
–0.1082561539 |
–0.1337576526 |
–0.1547658143 |
- |
- |
|
11 |
–0.0584280166 |
–0.0893884916 |
–0.1138959157 |
–0.1339980129 |
–0.1509377146 |
- |
|
12 |
–0.0434497186 |
–0.0735218839 |
–0.0972226862 |
–0.1165871475 |
–0.1328481031 |
–0.1467987268 |
|
13 |
–0.0306301563 |
–0.0599731110 |
–0.0830092969 |
–0.1017640813 |
–0.1174625474 |
–0.1308915697 |
|
14 |
–0.0195170409 |
–0.0482533788 |
–0.0707347568 |
–0.0889790893 |
–0.1042054207 |
–0.1171956771 |
|
15 |
–0.0097777104 |
–0.0380035757 |
–0.0600165553 |
–0.0778287325 |
–0.0926543857 |
–0.1052715064 |
|
16 |
–0.0011618546 |
–0.0289538058 |
–0.0505674185 |
–0.0680101578 |
–0.0824924974 |
–0.0947892654 |
|
17 |
0.0065226162 |
–0.0208972016 |
–0.0421673279 |
–0.0592915297 |
–0.0734772120 |
–0.0854966347 |
|
18 |
0.0134256679 |
–0.0136724619 |
–0.0346448650 |
–0.0514923184 |
–0.0654197288 |
–0.0771972516 |
|
19 |
0.0196661656 |
–0.0071518992 |
–0.0278644465 |
–0.0444698134 |
–0.0581708549 |
–0.0697359889 |
|
20 |
0.0253395974 |
–0.0012330702 |
–0.0217173864 |
–0.0381096814 |
–0.0516111124 |
–0.0629886511 |
|
q \ a |
12 |
13 |
14 |
15 |
16 |
17 |
|
13 |
–0.1425812024 |
- |
- |
- |
- |
- |
|
14 |
–0.1284758180 |
–0.1384135693 |
- |
- |
- |
- |
|
15 |
–0.1162027756 |
–0.1258131641 |
–0.1343658123 |
- |
- |
- |
|
16 |
–0.1054204902 |
–0.1147489005 |
–0.1230357282 |
–0.1304742530 |
- |
- |
|
17 |
–0.0958676592 |
–0.1049511732 |
–0.1130068333 |
–0.1202265633 |
–0.1267555301 |
- |
|
18 |
–0.0873409635 |
–0.0962102275 |
–0.1040633898 |
–0.1110912414 |
–0.1174379643 |
–0.1232147102 |
|
19 |
–0.0796798099 |
–0.0883604091 |
–0.0960350346 |
–0.1028935242 |
–0.1090792282 |
–0.1147025691 |
|
20 |
–0.0727556530 |
–0.0812691406 |
–0.0887854360 |
–0.0954935697 |
–0.1015361853 |
–0.1070230837 |
|
q \ a |
18 |
19 |
|
19 |
–0.1198500975 |
- |
|
20 |
–0.1120402662 |
–0.1166561329 |