L(s) = 1 | − 8·2-s + 18·3-s + 48·4-s − 49·5-s − 144·6-s − 105·7-s − 256·8-s + 243·9-s + 392·10-s + 725·11-s + 864·12-s − 56·13-s + 840·14-s − 882·15-s + 1.28e3·16-s + 1.52e3·17-s − 1.94e3·18-s − 722·19-s − 2.35e3·20-s − 1.89e3·21-s − 5.80e3·22-s + 1.70e3·23-s − 4.60e3·24-s − 3.42e3·25-s + 448·26-s + 2.91e3·27-s − 5.04e3·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.876·5-s − 1.63·6-s − 0.809·7-s − 1.41·8-s + 9-s + 1.23·10-s + 1.80·11-s + 1.73·12-s − 0.0919·13-s + 1.14·14-s − 1.01·15-s + 5/4·16-s + 1.27·17-s − 1.41·18-s − 0.458·19-s − 1.31·20-s − 0.935·21-s − 2.55·22-s + 0.670·23-s − 1.63·24-s − 1.09·25-s + 0.129·26-s + 0.769·27-s − 1.21·28-s + ⋯ |
Λ(s)=(=(12996s/2ΓC(s)2L(s)Λ(6−s)
Λ(s)=(=(12996s/2ΓC(s+5/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
12996
= 22⋅32⋅192
|
Sign: |
1
|
Analytic conductor: |
334.295 |
Root analytic conductor: |
4.27595 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 12996, ( :5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
2.213547065 |
L(21) |
≈ |
2.213547065 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1+p2T)2 |
| 3 | C1 | (1−p2T)2 |
| 19 | C1 | (1+p2T)2 |
good | 5 | D4 | 1+49T+5828T2+49p5T3+p10T4 |
| 7 | D4 | 1+15pT+10814T2+15p6T3+p10T4 |
| 11 | D4 | 1−725T+452486T2−725p5T3+p10T4 |
| 13 | D4 | 1+56T+334470T2+56p5T3+p10T4 |
| 17 | D4 | 1−1521T+1863232T2−1521p5T3+p10T4 |
| 23 | D4 | 1−1700T−160210T2−1700p5T3+p10T4 |
| 29 | D4 | 1−12724T+81235646T2−12724p5T3+p10T4 |
| 31 | D4 | 1−8558T+58292118T2−8558p5T3+p10T4 |
| 37 | D4 | 1−12434T+135627114T2−12434p5T3+p10T4 |
| 41 | D4 | 1−20230T+259503602T2−20230p5T3+p10T4 |
| 43 | D4 | 1−4895T+299556330T2−4895p5T3+p10T4 |
| 47 | D4 | 1−21059T+555084302T2−21059p5T3+p10T4 |
| 53 | D4 | 1−25196T+966248606T2−25196p5T3+p10T4 |
| 59 | D4 | 1−1560T+1429410214T2−1560p5T3+p10T4 |
| 61 | D4 | 1+2123T−588325956T2+2123p5T3+p10T4 |
| 67 | D4 | 1+46968T+3250701686T2+46968p5T3+p10T4 |
| 71 | D4 | 1−60056T+2271003086T2−60056p5T3+p10T4 |
| 73 | D4 | 1+57177T+4702746212T2+57177p5T3+p10T4 |
| 79 | D4 | 1+99368T+8517934254T2+99368p5T3+p10T4 |
| 83 | D4 | 1+21780T−3626701658T2+21780p5T3+p10T4 |
| 89 | D4 | 1+11856T+2415361198T2+11856p5T3+p10T4 |
| 97 | D4 | 1+77068T−559666266T2+77068p5T3+p10T4 |
show more | | |
show less | | |
L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−12.59077662442301502535017374809, −12.34699905643317480108195750971, −11.66689302118222171502422080708, −11.53877921073748755680088974846, −10.38404933270753762296681404857, −10.18348210530864781605951302574, −9.532086136105657560223432370297, −9.191725621198016247111052480499, −8.583966024772736185898767506845, −8.196511280388750974797120557625, −7.56581412496507055927749357261, −7.15622885548722860066789836077, −6.37570646886252990644723498936, −6.03295358005877565783144992649, −4.34245187045827182781694863796, −3.97500465481077572317087483570, −2.99102723988132078536658953077, −2.57445413174859067452073381954, −1.15250050606863482145280041400, −0.812121560386603378916570811270,
0.812121560386603378916570811270, 1.15250050606863482145280041400, 2.57445413174859067452073381954, 2.99102723988132078536658953077, 3.97500465481077572317087483570, 4.34245187045827182781694863796, 6.03295358005877565783144992649, 6.37570646886252990644723498936, 7.15622885548722860066789836077, 7.56581412496507055927749357261, 8.196511280388750974797120557625, 8.583966024772736185898767506845, 9.191725621198016247111052480499, 9.532086136105657560223432370297, 10.18348210530864781605951302574, 10.38404933270753762296681404857, 11.53877921073748755680088974846, 11.66689302118222171502422080708, 12.34699905643317480108195750971, 12.59077662442301502535017374809