L(s) = 1 | − 36·3-s − 8·5-s + 176·7-s + 810·9-s − 96·11-s − 676·13-s + 288·15-s + 1.22e3·17-s − 3.85e3·19-s − 6.33e3·21-s + 368·23-s − 3.27e3·25-s − 1.45e4·27-s + 264·29-s − 1.09e4·31-s + 3.45e3·33-s − 1.40e3·35-s + 1.20e4·37-s + 2.43e4·39-s + 3.27e4·41-s − 2.42e4·43-s − 6.48e3·45-s − 1.41e4·47-s + 2.34e3·49-s − 4.40e4·51-s + 3.73e4·53-s + 768·55-s + ⋯ |
L(s) = 1 | − 2.30·3-s − 0.143·5-s + 1.35·7-s + 10/3·9-s − 0.239·11-s − 1.10·13-s + 0.330·15-s + 1.02·17-s − 2.45·19-s − 3.13·21-s + 0.145·23-s − 1.04·25-s − 3.84·27-s + 0.0582·29-s − 2.04·31-s + 0.552·33-s − 0.194·35-s + 1.44·37-s + 2.56·39-s + 3.04·41-s − 2.00·43-s − 0.477·45-s − 0.936·47-s + 0.139·49-s − 2.37·51-s + 1.82·53-s + 0.0342·55-s + ⋯ |
Λ(s)=(=((216⋅34⋅134)s/2ΓC(s)4L(s)Λ(6−s)
Λ(s)=(=((216⋅34⋅134)s/2ΓC(s+5/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
216⋅34⋅134
|
Sign: |
1
|
Analytic conductor: |
1.00318×108 |
Root analytic conductor: |
10.0039 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 216⋅34⋅134, ( :5/2,5/2,5/2,5/2), 1)
|
Particular Values
L(3) |
= |
0 |
L(21) |
= |
0 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C1 | (1+p2T)4 |
| 13 | C1 | (1+p2T)4 |
good | 5 | C2≀S4 | 1+8T+668pT2−16808T3+5971334T4−16808p5T5+668p11T6+8p15T7+p20T8 |
| 7 | C2≀S4 | 1−176T+28628T2−3721296T3+605143094T4−3721296p5T5+28628p10T6−176p15T7+p20T8 |
| 11 | C2≀S4 | 1+96T+472148T2+3836512pT3+101801657814T4+3836512p6T5+472148p10T6+96p15T7+p20T8 |
| 17 | C2≀S4 | 1−72pT+4103356T2−4084243320T3+8525013382982T4−4084243320p5T5+4103356p10T6−72p16T7+p20T8 |
| 19 | C2≀S4 | 1+3856T+10968900T2+21938174576T3+37174174246918T4+21938174576p5T5+10968900p10T6+3856p15T7+p20T8 |
| 23 | C2≀S4 | 1−16pT+7998332T2+18196605392T3+21476708847014T4+18196605392p5T5+7998332p10T6−16p16T7+p20T8 |
| 29 | C2≀S4 | 1−264T+525052pT2+15901448616T3+746777890351734T4+15901448616p5T5+525052p11T6−264p15T7+p20T8 |
| 31 | C2≀S4 | 1+10928T+105783700T2+19405988912pT3+3743626337366998T4+19405988912p6T5+105783700p10T6+10928p15T7+p20T8 |
| 37 | C2≀S4 | 1−12008T+101704620T2−557784338872T3+5983242191546326T4−557784338872p5T5+101704620p10T6−12008p15T7+p20T8 |
| 41 | C2≀S4 | 1−32792T+727984380T2−11248602324648T3+139994437031392918T4−11248602324648p5T5+727984380p10T6−32792p15T7+p20T8 |
| 43 | C2≀S4 | 1+24288T+790918124T2+11423099410656T3+190216374004423830T4+11423099410656p5T5+790918124p10T6+24288p15T7+p20T8 |
| 47 | C2≀S4 | 1+14176T+735750212T2+8384297776032T3+239936046865012998T4+8384297776032p5T5+735750212p10T6+14176p15T7+p20T8 |
| 53 | C2≀S4 | 1−37384T+1240221324T2−18715686161688T3+444519628921633366T4−18715686161688p5T5+1240221324p10T6−37384p15T7+p20T8 |
| 59 | C2≀S4 | 1+11456T+1487883380T2−10207240326912T3+885850394336258646T4−10207240326912p5T5+1487883380p10T6+11456p15T7+p20T8 |
| 61 | C2≀S4 | 1+15752T+251537452T2+18907846324056T3+1392265203483209910T4+18907846324056p5T5+251537452p10T6+15752p15T7+p20T8 |
| 67 | C2≀S4 | 1−14032T+4075292836T2−37348416655344T3+7544579925679316358T4−37348416655344p5T5+4075292836p10T6−14032p15T7+p20T8 |
| 71 | C2≀S4 | 1+7552T+7061115716T2+39517413965888T3+18968128942170165926T4+39517413965888p5T5+7061115716p10T6+7552p15T7+p20T8 |
| 73 | C2≀S4 | 1−76104T+8970942652T2−453081798431416T3+28604127075938503974T4−453081798431416p5T5+8970942652p10T6−76104p15T7+p20T8 |
| 79 | C2≀S4 | 1+61056T+9407034300T2+512811277886592T3+40567621248160124998T4+512811277886592p5T5+9407034300p10T6+61056p15T7+p20T8 |
| 83 | C2≀S4 | 1+233856T+31172189780T2+2964896486052928T3+21⋯78T4+2964896486052928p5T5+31172189780p10T6+233856p15T7+p20T8 |
| 89 | C2≀S4 | 1−86136T+15628720700T2−1055365130959432T3+12⋯98T4−1055365130959432p5T5+15628720700p10T6−86136p15T7+p20T8 |
| 97 | C2≀S4 | 1−34664T+30099720028T2−766306396866008T3+36⋯18T4−766306396866008p5T5+30099720028p10T6−34664p15T7+p20T8 |
show more | | |
show less | | |
L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−7.41207973736681211725766963521, −7.03398761106698099477055053103, −6.60156698256176312986534596940, −6.54808797305051215125570994080, −6.51010176781368744890482943552, −5.93542562937579872222415942976, −5.84502561017796013158071244753, −5.60378161056747434002530798638, −5.45210237742259553200626864292, −5.24320583567421173657447011017, −4.79402840144759206561296732718, −4.74733949621629371010678213133, −4.59436972118887523517682051154, −4.18412220096567879177257220488, −3.94064024044333466250636926016, −3.70081002582961160690617050603, −3.69738870379290766701756430400, −2.76035962454376491693048972960, −2.62519220187063951380519140781, −2.31381533049260790471630204992, −2.14437592851221259758117633117, −1.51469152138437644897254001020, −1.47443454240362464698813516608, −1.04091468871633291076103873305, −1.02548064601404728500736125273, 0, 0, 0, 0,
1.02548064601404728500736125273, 1.04091468871633291076103873305, 1.47443454240362464698813516608, 1.51469152138437644897254001020, 2.14437592851221259758117633117, 2.31381533049260790471630204992, 2.62519220187063951380519140781, 2.76035962454376491693048972960, 3.69738870379290766701756430400, 3.70081002582961160690617050603, 3.94064024044333466250636926016, 4.18412220096567879177257220488, 4.59436972118887523517682051154, 4.74733949621629371010678213133, 4.79402840144759206561296732718, 5.24320583567421173657447011017, 5.45210237742259553200626864292, 5.60378161056747434002530798638, 5.84502561017796013158071244753, 5.93542562937579872222415942976, 6.51010176781368744890482943552, 6.54808797305051215125570994080, 6.60156698256176312986534596940, 7.03398761106698099477055053103, 7.41207973736681211725766963521