L(s) = 1 | + 11·3-s − 31·5-s + 39·7-s − 418·9-s − 340·11-s + 676·13-s − 341·15-s + 1.75e3·17-s − 2.00e3·19-s + 429·21-s + 6.79e3·23-s − 4.19e3·25-s − 9.71e3·27-s − 6.76e3·29-s + 1.86e4·31-s − 3.74e3·33-s − 1.20e3·35-s + 573·37-s + 7.43e3·39-s + 1.89e3·41-s − 2.83e4·43-s + 1.29e4·45-s + 2.16e4·47-s − 3.55e4·49-s + 1.93e4·51-s + 4.59e4·53-s + 1.05e4·55-s + ⋯ |
L(s) = 1 | + 0.705·3-s − 0.554·5-s + 0.300·7-s − 1.72·9-s − 0.847·11-s + 1.10·13-s − 0.391·15-s + 1.47·17-s − 1.27·19-s + 0.212·21-s + 2.67·23-s − 1.34·25-s − 2.56·27-s − 1.49·29-s + 3.48·31-s − 0.597·33-s − 0.166·35-s + 0.0688·37-s + 0.782·39-s + 0.175·41-s − 2.33·43-s + 0.953·45-s + 1.42·47-s − 2.11·49-s + 1.04·51-s + 2.24·53-s + 0.469·55-s + ⋯ |
Λ(s)=(=((224⋅134)s/2ΓC(s)4L(s)Λ(6−s)
Λ(s)=(=((224⋅134)s/2ΓC(s+5/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
224⋅134
|
Sign: |
1
|
Analytic conductor: |
3.17055×108 |
Root analytic conductor: |
11.5515 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 224⋅134, ( :5/2,5/2,5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
6.711708024 |
L(21) |
≈ |
6.711708024 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 13 | C1 | (1−p2T)4 |
good | 3 | C2≀S4 | 1−11T+539T2−808T3+39808pT4−808p5T5+539p10T6−11p15T7+p20T8 |
| 5 | C2≀S4 | 1+31T+5153T2+252422T3+22858118T4+252422p5T5+5153p10T6+31p15T7+p20T8 |
| 7 | C2≀S4 | 1−39T+37057T2+1509244T3+574546398T4+1509244p5T5+37057p10T6−39p15T7+p20T8 |
| 11 | C2≀S4 | 1+340T+127772T2−3699508pT3−2539680526pT4−3699508p6T5+127772p10T6+340p15T7+p20T8 |
| 17 | C2≀S4 | 1−1757T+3521321T2−4384174254T3+7201578383694T4−4384174254p5T5+3521321p10T6−1757p15T7+p20T8 |
| 19 | C2≀S4 | 1+2000T+1315204T2+5123411728T3+16378456307222T4+5123411728p5T5+1315204p10T6+2000p15T7+p20T8 |
| 23 | C2≀S4 | 1−6796T+37313132T2−124874503804T3+376774335025606T4−124874503804p5T5+37313132p10T6−6796p15T7+p20T8 |
| 29 | C2≀S4 | 1+6768T+44087996T2+145395725328T3+820460284261782T4+145395725328p5T5+44087996p10T6+6768p15T7+p20T8 |
| 31 | C2≀S4 | 1−18638T+155809588T2−751376049398T3+3287012745889942T4−751376049398p5T5+155809588p10T6−18638p15T7+p20T8 |
| 37 | C2≀S4 | 1−573T+239640937T2−202243860778T3+23587529015810070T4−202243860778p5T5+239640937p10T6−573p15T7+p20T8 |
| 41 | C2≀S4 | 1−1890T+311989748T2−958041784790T3+46445767927334790T4−958041784790p5T5+311989748p10T6−1890p15T7+p20T8 |
| 43 | C2≀S4 | 1+28327T+616834087T2+7694590240592T3+104664597506136140T4+7694590240592p5T5+616834087p10T6+28327p15T7+p20T8 |
| 47 | C2≀S4 | 1−21603T+666176561T2−10471292070516T3+201019507301163414T4−10471292070516p5T5+666176561p10T6−21603p15T7+p20T8 |
| 53 | C2≀S4 | 1−45950T+1702757516T2−47289854346690T3+1021571292731542854T4−47289854346690p5T5+1702757516p10T6−45950p15T7+p20T8 |
| 59 | C2≀S4 | 1+916T+2171389052T2−8148340134204T3+2041156776912096870T4−8148340134204p5T5+2171389052p10T6+916p15T7+p20T8 |
| 61 | C2≀S4 | 1−55730T+67085956pT2−139009698415134T3+5445275958006059382T4−139009698415134p5T5+67085956p11T6−55730p15T7+p20T8 |
| 67 | C2≀S4 | 1+7736T+2054018164T2+34631039929752T3+3527055037627735350T4+34631039929752p5T5+2054018164p10T6+7736p15T7+p20T8 |
| 71 | C2≀S4 | 1−25229T+5569425785T2−91697921781076T3+13582162611069378350T4−91697921781076p5T5+5569425785p10T6−25229p15T7+p20T8 |
| 73 | C2≀S4 | 1+82484T+3610412740T2−27086603654356T3−4146949427050770410T4−27086603654356p5T5+3610412740p10T6+82484p15T7+p20T8 |
| 79 | C2≀S4 | 1−4936T+3136547068T2−276560357630760T3+2205991352982184134T4−276560357630760p5T5+3136547068p10T6−4936p15T7+p20T8 |
| 83 | C2≀S4 | 1−119782T+16058583212T2−1197963265006102T3+91320962773018703350T4−1197963265006102p5T5+16058583212p10T6−119782p15T7+p20T8 |
| 89 | C2≀S4 | 1+153492T+23233599524T2+2201629256787724T3+20⋯50T4+2201629256787724p5T5+23233599524p10T6+153492p15T7+p20T8 |
| 97 | C2≀S4 | 1−13364T+15673117972T2+697855089119668T3+10⋯54T4+697855089119668p5T5+15673117972p10T6−13364p15T7+p20T8 |
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L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−6.68452944058160745639535847591, −6.21388352156540350942989652364, −5.96396305695149126701933393067, −5.86004992147183374143238608691, −5.74046864824556768238284674250, −5.31105978721251068369414692196, −5.23585697986123428123098977581, −4.95571247128615953641857023079, −4.77679496844087350546108824602, −4.27494296219800578860774614628, −4.08124675641141145590914122318, −3.90391109438156301003332171714, −3.68053775398656905460239990908, −3.17490520488419122908248086056, −3.08766452296930958295682149124, −2.93715948922570929185706773372, −2.80787044476721731047360195841, −2.33137836983128159142152636132, −1.98851508988812689578286299673, −1.86003840601409194800642001748, −1.42673027344858036112227733189, −1.08720116971035349170994543447, −0.61079487544616993458609765486, −0.54774658151306755564396219249, −0.31418994809951318271021889752,
0.31418994809951318271021889752, 0.54774658151306755564396219249, 0.61079487544616993458609765486, 1.08720116971035349170994543447, 1.42673027344858036112227733189, 1.86003840601409194800642001748, 1.98851508988812689578286299673, 2.33137836983128159142152636132, 2.80787044476721731047360195841, 2.93715948922570929185706773372, 3.08766452296930958295682149124, 3.17490520488419122908248086056, 3.68053775398656905460239990908, 3.90391109438156301003332171714, 4.08124675641141145590914122318, 4.27494296219800578860774614628, 4.77679496844087350546108824602, 4.95571247128615953641857023079, 5.23585697986123428123098977581, 5.31105978721251068369414692196, 5.74046864824556768238284674250, 5.86004992147183374143238608691, 5.96396305695149126701933393067, 6.21388352156540350942989652364, 6.68452944058160745639535847591