L(s) = 1 | − 6·3-s + 18·5-s − 10·7-s + 27·9-s − 4·11-s − 26·13-s − 108·15-s − 52·17-s + 142·19-s + 60·21-s − 280·23-s + 10·25-s − 108·27-s + 424·29-s − 178·31-s + 24·33-s − 180·35-s − 136·37-s + 156·39-s − 226·41-s + 72·43-s + 486·45-s + 176·47-s − 186·49-s + 312·51-s + 788·53-s − 72·55-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1.60·5-s − 0.539·7-s + 9-s − 0.109·11-s − 0.554·13-s − 1.85·15-s − 0.741·17-s + 1.71·19-s + 0.623·21-s − 2.53·23-s + 2/25·25-s − 0.769·27-s + 2.71·29-s − 1.03·31-s + 0.126·33-s − 0.869·35-s − 0.604·37-s + 0.640·39-s − 0.860·41-s + 0.255·43-s + 1.60·45-s + 0.546·47-s − 0.542·49-s + 0.856·51-s + 2.04·53-s − 0.176·55-s + ⋯ |
Λ(s)=(=(6230016s/2ΓC(s)2L(s)Λ(4−s)
Λ(s)=(=(6230016s/2ΓC(s+3/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
6230016
= 212⋅32⋅132
|
Sign: |
1
|
Analytic conductor: |
21688.0 |
Root analytic conductor: |
12.1354 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 6230016, ( :3/2,3/2), 1)
|
Particular Values
L(2) |
≈ |
0.7916360677 |
L(21) |
≈ |
0.7916360677 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C1 | (1+pT)2 |
| 13 | C1 | (1+pT)2 |
good | 5 | D4 | 1−18T+314T2−18p3T3+p6T4 |
| 7 | D4 | 1+10T+286T2+10p3T3+p6T4 |
| 11 | D4 | 1+4T−666T2+4p3T3+p6T4 |
| 17 | D4 | 1+52T+8054T2+52p3T3+p6T4 |
| 19 | D4 | 1−142T+16702T2−142p3T3+p6T4 |
| 23 | D4 | 1+280T+41486T2+280p3T3+p6T4 |
| 29 | D4 | 1−424T+93110T2−424p3T3+p6T4 |
| 31 | D4 | 1+178T+48990T2+178p3T3+p6T4 |
| 37 | D4 | 1+136T+56358T2+136p3T3+p6T4 |
| 41 | D4 | 1+226T+144474T2+226p3T3+p6T4 |
| 43 | D4 | 1−72T+28662T2−72p3T3+p6T4 |
| 47 | D4 | 1−176T−29410T2−176p3T3+p6T4 |
| 53 | D4 | 1−788T+451902T2−788p3T3+p6T4 |
| 59 | D4 | 1+728T+542166T2+728p3T3+p6T4 |
| 61 | D4 | 1−736T+564838T2−736p3T3+p6T4 |
| 67 | D4 | 1+1054T+684622T2+1054p3T3+p6T4 |
| 71 | D4 | 1+660T+721294T2+660p3T3+p6T4 |
| 73 | D4 | 1−24T−59650T2−24p3T3+p6T4 |
| 79 | D4 | 1−40T−308514T2−40p3T3+p6T4 |
| 83 | D4 | 1+1344T+1516550T2+1344p3T3+p6T4 |
| 89 | D4 | 1−314T+619250T2−314p3T3+p6T4 |
| 97 | D4 | 1+264T+1841070T2+264p3T3+p6T4 |
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
−8.663040339537159996800988353922, −8.637322623139489185634022187790, −7.77389835717044415361569491672, −7.65169715757925005453971726819, −6.94948595431688625402849838888, −6.88645624154527224327976326832, −6.22274566693002830088751709379, −6.04468920200788309190695031104, −5.70595912021884655323917946443, −5.46684814163672059634228752514, −4.87653997649164264083503781619, −4.61273054487641761704473496898, −3.99933366716651493691274307760, −3.60574065369681497554899255004, −2.81771518075726149318426536456, −2.53772379526104580160829602345, −1.82099928139287013288395700752, −1.62939953893538085647119128250, −0.893979176437509802330674656599, −0.20963803074606327814336657667,
0.20963803074606327814336657667, 0.893979176437509802330674656599, 1.62939953893538085647119128250, 1.82099928139287013288395700752, 2.53772379526104580160829602345, 2.81771518075726149318426536456, 3.60574065369681497554899255004, 3.99933366716651493691274307760, 4.61273054487641761704473496898, 4.87653997649164264083503781619, 5.46684814163672059634228752514, 5.70595912021884655323917946443, 6.04468920200788309190695031104, 6.22274566693002830088751709379, 6.88645624154527224327976326832, 6.94948595431688625402849838888, 7.65169715757925005453971726819, 7.77389835717044415361569491672, 8.637322623139489185634022187790, 8.663040339537159996800988353922