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) |
≈ |
52.19958565 |
L(21) |
≈ |
52.19958565 |
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 |
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
−7.18102493425257521845906584047, −6.69808121762116658713841796084, −6.36687338852324465801638260491, −6.36598688716242558748899118855, −6.07676048605494948911704554864, −5.65305201763409090172770186779, −5.28771184342674708876054141718, −5.24428039898948773581046069210, −5.03250606626888387137054384718, −4.25695826403004166418053211987, −4.16156170581148714606795259175, −4.07052994801156078545224181445, −4.06828331719991921553191082911, −3.42814849294869845924327700258, −3.28893880922404834078714316081, −3.07503857938357793721555985406, −3.04909450570473859424203583454, −2.11586744594484482600895167391, −2.09448299306356188416019700470, −2.05406851904030609848654173438, −1.89835545859806929760184414904, −1.14794996546507994633999169744, −0.879963474089920637508027276258, −0.64547434166418236436075483797, −0.62230237170248182907945090259,
0.62230237170248182907945090259, 0.64547434166418236436075483797, 0.879963474089920637508027276258, 1.14794996546507994633999169744, 1.89835545859806929760184414904, 2.05406851904030609848654173438, 2.09448299306356188416019700470, 2.11586744594484482600895167391, 3.04909450570473859424203583454, 3.07503857938357793721555985406, 3.28893880922404834078714316081, 3.42814849294869845924327700258, 4.06828331719991921553191082911, 4.07052994801156078545224181445, 4.16156170581148714606795259175, 4.25695826403004166418053211987, 5.03250606626888387137054384718, 5.24428039898948773581046069210, 5.28771184342674708876054141718, 5.65305201763409090172770186779, 6.07676048605494948911704554864, 6.36598688716242558748899118855, 6.36687338852324465801638260491, 6.69808121762116658713841796084, 7.18102493425257521845906584047