L(s) = 1 | + 15·5-s + 14·7-s + 22·11-s + 8·13-s + 34·17-s − 4·19-s + 176·23-s + 150·25-s − 98·29-s + 88·31-s + 210·35-s + 284·37-s + 8·41-s + 504·43-s + 280·47-s − 621·49-s + 150·53-s + 330·55-s + 350·59-s + 350·61-s + 120·65-s + 804·67-s + 500·71-s − 486·73-s + 308·77-s + 1.59e3·79-s − 684·83-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.755·7-s + 0.603·11-s + 0.170·13-s + 0.485·17-s − 0.0482·19-s + 1.59·23-s + 6/5·25-s − 0.627·29-s + 0.509·31-s + 1.01·35-s + 1.26·37-s + 0.0304·41-s + 1.78·43-s + 0.868·47-s − 1.81·49-s + 0.388·53-s + 0.809·55-s + 0.772·59-s + 0.734·61-s + 0.228·65-s + 1.46·67-s + 0.835·71-s − 0.779·73-s + 0.455·77-s + 2.26·79-s − 0.904·83-s + ⋯ |
Λ(s)=(=((215⋅36⋅53)s/2ΓC(s)3L(s)Λ(4−s)
Λ(s)=(=((215⋅36⋅53)s/2ΓC(s+3/2)3L(s)Λ(1−s)
Degree: |
6 |
Conductor: |
215⋅36⋅53
|
Sign: |
1
|
Analytic conductor: |
613317. |
Root analytic conductor: |
9.21752 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(6, 215⋅36⋅53, ( :3/2,3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
15.15029529 |
L(21) |
≈ |
15.15029529 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | | 1 |
| 5 | C1 | (1−pT)3 |
good | 7 | S4×C2 | 1−2pT+817T2−9196T3+817p3T4−2p7T5+p9T6 |
| 11 | S4×C2 | 1−2pT+2149T2−69196T3+2149p3T4−2p7T5+p9T6 |
| 13 | S4×C2 | 1−8T+2127T2−119632T3+2127p3T4−8p6T5+p9T6 |
| 17 | S4×C2 | 1−2pT+7951T2−430972T3+7951p3T4−2p7T5+p9T6 |
| 19 | S4×C2 | 1+4T+7649T2+529240T3+7649p3T4+4p6T5+p9T6 |
| 23 | S4×C2 | 1−176T+35205T2−4252064T3+35205p3T4−176p6T5+p9T6 |
| 29 | S4×C2 | 1+98T+24635T2+7058028T3+24635p3T4+98p6T5+p9T6 |
| 31 | S4×C2 | 1−88T+52685T2−3535312T3+52685p3T4−88p6T5+p9T6 |
| 37 | S4×C2 | 1−284T+49559T2−13028696T3+49559p3T4−284p6T5+p9T6 |
| 41 | S4×C2 | 1−8T+121451T2−10434960T3+121451p3T4−8p6T5+p9T6 |
| 43 | S4×C2 | 1−504T+174009T2−40942288T3+174009p3T4−504p6T5+p9T6 |
| 47 | S4×C2 | 1−280T+115997T2−54406224T3+115997p3T4−280p6T5+p9T6 |
| 53 | S4×C2 | 1−150T+253683T2−44718980T3+253683p3T4−150p6T5+p9T6 |
| 59 | S4×C2 | 1−350T+202005T2+17161444T3+202005p3T4−350p6T5+p9T6 |
| 61 | S4×C2 | 1−350T+204635T2−134954132T3+204635p3T4−350p6T5+p9T6 |
| 67 | S4×C2 | 1−12pT+1077441T2−489158232T3+1077441p3T4−12p7T5+p9T6 |
| 71 | S4×C2 | 1−500T+1072245T2−353953688T3+1072245p3T4−500p6T5+p9T6 |
| 73 | S4×C2 | 1+486T+1084311T2+377044724T3+1084311p3T4+486p6T5+p9T6 |
| 79 | S4×C2 | 1−1592T+2161789T2−1645964560T3+2161789p3T4−1592p6T5+p9T6 |
| 83 | S4×C2 | 1+684T+1381713T2+561532168T3+1381713p3T4+684p6T5+p9T6 |
| 89 | S4×C2 | 1−668T+2135307T2−937072184T3+2135307p3T4−668p6T5+p9T6 |
| 97 | S4×C2 | 1+1394T+2868623T2+2439375164T3+2868623p3T4+1394p6T5+p9T6 |
show more | | |
show less | | |
L(s)=p∏ j=1∏6(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.327354213251203685812491872786, −7.66989436275914303851749250651, −7.56927378934543411240532753381, −7.40572822470370450200737239686, −6.73298807393961043426792631396, −6.73060017982676221252337766656, −6.65226096481712868977592850336, −5.89073612469584847112310058556, −5.84036727135581625349582303971, −5.83775254906170227105516882193, −5.11527604403730804942307575895, −5.00067207131618732558532909073, −4.97389523542831345951528206394, −4.21224768658033662536849521249, −4.15528441469696365237606685948, −3.87034662897963979607725059497, −3.18621961135979935708094206377, −2.99808152661762629957099906458, −2.78005104036952292404722075943, −2.15220079774395696503825157241, −1.84756298653822527608429440287, −1.79244550709812595813468615436, −0.966778151300225403399521368991, −0.818345420903198442924507445162, −0.64710079732116236320918425889,
0.64710079732116236320918425889, 0.818345420903198442924507445162, 0.966778151300225403399521368991, 1.79244550709812595813468615436, 1.84756298653822527608429440287, 2.15220079774395696503825157241, 2.78005104036952292404722075943, 2.99808152661762629957099906458, 3.18621961135979935708094206377, 3.87034662897963979607725059497, 4.15528441469696365237606685948, 4.21224768658033662536849521249, 4.97389523542831345951528206394, 5.00067207131618732558532909073, 5.11527604403730804942307575895, 5.83775254906170227105516882193, 5.84036727135581625349582303971, 5.89073612469584847112310058556, 6.65226096481712868977592850336, 6.73060017982676221252337766656, 6.73298807393961043426792631396, 7.40572822470370450200737239686, 7.56927378934543411240532753381, 7.66989436275914303851749250651, 8.327354213251203685812491872786