L(s) = 1 | − 36·3-s + 810·9-s + 462·11-s − 602·13-s − 228·17-s − 358·19-s + 2.14e3·23-s − 3.52e3·25-s − 1.45e4·27-s − 5.53e3·29-s − 830·31-s − 1.66e4·33-s + 3.91e3·37-s + 2.16e4·39-s − 8.31e3·41-s − 1.45e4·43-s − 4.17e4·47-s + 8.20e3·51-s − 2.21e4·53-s + 1.28e4·57-s − 3.28e4·59-s − 8.37e4·61-s + 8.00e4·67-s − 7.73e4·69-s + 4.47e4·71-s + 2.24e4·73-s + 1.26e5·75-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 10/3·9-s + 1.15·11-s − 0.987·13-s − 0.191·17-s − 0.227·19-s + 0.846·23-s − 1.12·25-s − 3.84·27-s − 1.22·29-s − 0.155·31-s − 2.65·33-s + 0.470·37-s + 2.28·39-s − 0.772·41-s − 1.19·43-s − 2.75·47-s + 0.441·51-s − 1.08·53-s + 0.525·57-s − 1.22·59-s − 2.88·61-s + 2.17·67-s − 1.95·69-s + 1.05·71-s + 0.493·73-s + 2.60·75-s + ⋯ |
Λ(s)=(=((28⋅34⋅78)s/2ΓC(s)4L(s)Λ(6−s)
Λ(s)=(=((28⋅34⋅78)s/2ΓC(s+5/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
28⋅34⋅78
|
Sign: |
1
|
Analytic conductor: |
7.90954×107 |
Root analytic conductor: |
9.71111 |
Motivic weight: |
5 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 28⋅34⋅78, ( :5/2,5/2,5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
0.04641101262 |
L(21) |
≈ |
0.04641101262 |
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 |
| 7 | | 1 |
good | 5 | C2≀S4 | 1+3523T2+132p4T3+13251176T4+132p9T5+3523p10T6+p20T8 |
| 11 | C2≀S4 | 1−42pT+268705T2−50129610T3+18647530376T4−50129610p5T5+268705p10T6−42p16T7+p20T8 |
| 13 | C2≀S4 | 1+602T−12679pT2+145984794T3+357680241104T4+145984794p5T5−12679p11T6+602p15T7+p20T8 |
| 17 | C2≀S4 | 1+228T+3869284T2+1347363564T3+6940431119366T4+1347363564p5T5+3869284p10T6+228p15T7+p20T8 |
| 19 | C2≀S4 | 1+358T+7365277T2+2785970626T3+25078082046268T4+2785970626p5T5+7365277p10T6+358p15T7+p20T8 |
| 23 | C2≀S4 | 1−2148T+16852828T2−16061095668T3+120569983154918T4−16061095668p5T5+16852828p10T6−2148p15T7+p20T8 |
| 29 | C2≀S4 | 1+5532T+56710687T2+131401968540T3+1151610033241988T4+131401968540p5T5+56710687p10T6+5532p15T7+p20T8 |
| 31 | C2≀S4 | 1+830T+31255592T2−217237417272T3+250607576745089T4−217237417272p5T5+31255592p10T6+830p15T7+p20T8 |
| 37 | C2≀S4 | 1−3914T+81854461T2−709188215618T3+5749597008366352T4−709188215618p5T5+81854461p10T6−3914p15T7+p20T8 |
| 41 | C2≀S4 | 1+8316T+131719112T2−1793338464780T3−11507939827736466T4−1793338464780p5T5+131719112p10T6+8316p15T7+p20T8 |
| 43 | C2≀S4 | 1+14518T+553805833T2+5594375858722T3+119245786138973608T4+5594375858722p5T5+553805833p10T6+14518p15T7+p20T8 |
| 47 | C2≀S4 | 1+41700T+1201404476T2+465323341500pT3+369215501257269942T4+465323341500p6T5+1201404476p10T6+41700p15T7+p20T8 |
| 53 | C2≀S4 | 1+22164T+612500711T2−9154492772292T3−135301388975352444T4−9154492772292p5T5+612500711p10T6+22164p15T7+p20T8 |
| 59 | C2≀S4 | 1+32886T+2165272693T2+53279788989366T3+2201676784894672940T4+53279788989366p5T5+2165272693p10T6+32886p15T7+p20T8 |
| 61 | C2≀S4 | 1+83732T+5138847044T2+213873087755964T3+7191474746319250502T4+213873087755964p5T5+5138847044p10T6+83732p15T7+p20T8 |
| 67 | C2≀S4 | 1−80034T+6110296361T2−278312842674138T3+12284801359836541152T4−278312842674138p5T5+6110296361p10T6−80034p15T7+p20T8 |
| 71 | C2≀S4 | 1−44772T+5603702956T2−219545846683812T3+14130153190557486902T4−219545846683812p5T5+5603702956p10T6−44772p15T7+p20T8 |
| 73 | C2≀S4 | 1−22470T+3732785609T2−14345610615750T3+8330122733588006676T4−14345610615750p5T5+3732785609p10T6−22470p15T7+p20T8 |
| 79 | C2≀S4 | 1−75286T+10257886376T2−522030297673368T3+42333862433009559377T4−522030297673368p5T5+10257886376p10T6−75286p15T7+p20T8 |
| 83 | C2≀S4 | 1+17418T+11662792993T2+252243184391142T3+61381737302005449608T4+252243184391142p5T5+11662792993p10T6+17418p15T7+p20T8 |
| 89 | C2≀S4 | 1+28944T+4985253824T2+135699519340656T3−6573734910358628322T4+135699519340656p5T5+4985253824p10T6+28944p15T7+p20T8 |
| 97 | C2≀S4 | 1+216678T+43602999305T2+4974916579453302T3+56⋯00T4+4974916579453302p5T5+43602999305p10T6+216678p15T7+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
−6.80631305418479735947714268971, −6.44844864753482093055307576236, −6.42091176549015319922998194470, −6.26383618306861954487667524131, −6.09756040032114426162042649551, −5.57831951285068741473951625650, −5.32469718375663487150801739064, −5.30618578983063020268865576510, −5.10220746236652971520742976048, −4.68324149007996098994271096569, −4.38271679315724081885495903192, −4.32774520428248031298180763401, −4.23858047685534849350317826326, −3.44234435551543409629463844544, −3.43523010573100753682072297173, −3.20610846984594359584882355082, −2.93010157898980910543318897937, −2.07640048635663732507483068213, −1.91963735283839528513543943716, −1.78907495028052741552925999890, −1.61516258369381417752183062074, −1.05591681552149206355491521692, −0.60987739235975743285295035149, −0.56103016638208437468744430182, −0.04074361905608839665991496910,
0.04074361905608839665991496910, 0.56103016638208437468744430182, 0.60987739235975743285295035149, 1.05591681552149206355491521692, 1.61516258369381417752183062074, 1.78907495028052741552925999890, 1.91963735283839528513543943716, 2.07640048635663732507483068213, 2.93010157898980910543318897937, 3.20610846984594359584882355082, 3.43523010573100753682072297173, 3.44234435551543409629463844544, 4.23858047685534849350317826326, 4.32774520428248031298180763401, 4.38271679315724081885495903192, 4.68324149007996098994271096569, 5.10220746236652971520742976048, 5.30618578983063020268865576510, 5.32469718375663487150801739064, 5.57831951285068741473951625650, 6.09756040032114426162042649551, 6.26383618306861954487667524131, 6.42091176549015319922998194470, 6.44844864753482093055307576236, 6.80631305418479735947714268971