L(s) = 1 | − 14·3-s + 42·5-s + 894·9-s − 7.42e3·11-s − 2.36e4·13-s − 588·15-s + 1.57e4·17-s + 2.66e4·19-s − 3.26e4·23-s + 1.24e5·25-s + 2.64e4·27-s − 3.16e5·29-s − 1.80e5·31-s + 1.03e5·33-s + 4.58e4·37-s + 3.31e5·39-s + 6.43e5·41-s + 2.04e6·43-s + 3.75e4·45-s + 1.66e6·47-s − 2.21e5·51-s + 4.10e5·53-s − 3.11e5·55-s − 3.72e5·57-s + 1.70e6·59-s − 5.47e5·61-s − 9.93e5·65-s + ⋯ |
L(s) = 1 | − 0.299·3-s + 0.150·5-s + 0.408·9-s − 1.68·11-s − 2.98·13-s − 0.0449·15-s + 0.779·17-s + 0.890·19-s − 0.559·23-s + 1.59·25-s + 0.258·27-s − 2.40·29-s − 1.08·31-s + 0.503·33-s + 0.148·37-s + 0.894·39-s + 1.45·41-s + 3.92·43-s + 0.0614·45-s + 2.34·47-s − 0.233·51-s + 0.378·53-s − 0.252·55-s − 0.266·57-s + 1.07·59-s − 0.308·61-s − 0.448·65-s + ⋯ |
Λ(s)=(=((28⋅78)s/2ΓC(s)4L(s)Λ(8−s)
Λ(s)=(=((28⋅78)s/2ΓC(s+7/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
28⋅78
|
Sign: |
1
|
Analytic conductor: |
1.40535×107 |
Root analytic conductor: |
7.82479 |
Motivic weight: |
7 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 28⋅78, ( :7/2,7/2,7/2,7/2), 1)
|
Particular Values
L(4) |
≈ |
2.903552649 |
L(21) |
≈ |
2.903552649 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 7 | | 1 |
good | 3 | D4×C2 | 1+14T−698T2−16240pT3−490265p2T4−16240p8T5−698p14T6+14p21T7+p28T8 |
| 5 | D4×C2 | 1−42T−123166T2+263088pT3+374647071p2T4+263088p8T5−123166p14T6−42p21T7+p28T8 |
| 11 | D4×C2 | 1+7428T+8632202T2+56219857920T3+711291094304619T4+56219857920p7T5+8632202p14T6+7428p21T7+p28T8 |
| 13 | D4 | (1+70p2T+160198410T2+70p9T3+p14T4)2 |
| 17 | D4×C2 | 1−15792T−16280606pT2+4651056365760T3+6130718726782755T4+4651056365760p7T5−16280606p15T6−15792p21T7+p28T8 |
| 19 | D4×C2 | 1−26614T−1253942090T2−4644239023312T3+2418273122215699231T4−4644239023312p7T5−1253942090p14T6−26614p21T7+p28T8 |
| 23 | D4×C2 | 1+32640T+2057178482T2−254639647088640T3−14236548990664871085T4−254639647088640p7T5+2057178482p14T6+32640p21T7+p28T8 |
| 29 | D4 | (1+158016T+39988772806T2+158016p7T3+p14T4)2 |
| 31 | D4×C2 | 1+180740T−5817373358T2−2989603578895360T3−17⋯57T4−2989603578895360p7T5−5817373358p14T6+180740p21T7+p28T8 |
| 37 | D4×C2 | 1−45824T+20185374250T2+9529068243880960T3−88⋯21T4+9529068243880960p7T5+20185374250p14T6−45824p21T7+p28T8 |
| 41 | D4 | (1−321720T+181439440606T2−321720p7T3+p14T4)2 |
| 43 | D4 | (1−1023868T+671194246566T2−1023868p7T3+p14T4)2 |
| 47 | D4×C2 | 1−1665972T+1099984870706T2−1103259291706623744T3+11⋯23T4−1103259291706623744p7T5+1099984870706p14T6−1665972p21T7+p28T8 |
| 53 | D4×C2 | 1−410628T−2165589726190T2+6248608032034800T3+38⋯99T4+6248608032034800p7T5−2165589726190p14T6−410628p21T7+p28T8 |
| 59 | D4×C2 | 1−1702134T−1128927975562T2+1618924907272816080T3+28⋯99T4+1618924907272816080p7T5−1128927975562p14T6−1702134p21T7+p28T8 |
| 61 | D4×C2 | 1+547526T−5309538579326T2−370216478913573040T3+20⋯67T4−370216478913573040p7T5−5309538579326p14T6+547526p21T7+p28T8 |
| 67 | D4×C2 | 1−2590616T+2056450789210T2+19343048712604086400T3−55⋯01T4+19343048712604086400p7T5+2056450789210p14T6−2590616p21T7+p28T8 |
| 71 | D4 | (1−4129272T+22218672158062T2−4129272p7T3+p14T4)2 |
| 73 | D4×C2 | 1+8008868T+26118740970730T2+1747516196782370000pT3+11⋯31p2T4+1747516196782370000p8T5+26118740970730p14T6+8008868p21T7+p28T8 |
| 79 | D4×C2 | 1+2470456T+19757905667170T2−12⋯12T3−29⋯49T4−12⋯12p7T5+19757905667170p14T6+2470456p21T7+p28T8 |
| 83 | D4 | (1−9900786T+68835957963214T2−9900786p7T3+p14T4)2 |
| 89 | D4×C2 | 1−15423492T+94566779700986T2−84⋯40T3+80⋯87T4−84⋯40p7T5+94566779700986p14T6−15423492p21T7+p28T8 |
| 97 | D4 | (1−17377472T+164552259333822T2−17377472p7T3+p14T4)2 |
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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
−7.70104550776557944944898344050, −7.40037343241422406961754195505, −7.30902306289940021185878171534, −7.28985246771308818698677641825, −7.03503218981262130220260253290, −6.18874456559915966245969273110, −6.10365673643291093943405340781, −5.78943584648158753039347791396, −5.57364347261514177000946544966, −5.28528784141347941295761509725, −4.87053457832152162721405372955, −4.75400070048858678017954387518, −4.72504937983569954058748158577, −4.00333840680333740177532797361, −3.73243684450409317502394283715, −3.43063295030872095826499338234, −3.07598752514072215909753051672, −2.46253907578939283267205469826, −2.25810018513896790548440165901, −2.21423970157092315197588024821, −2.11830791649777830270543068506, −0.977529502770832894696557062749, −0.842447520547978161789360708025, −0.68671340316128552120876657741, −0.24913077485483568455998933811,
0.24913077485483568455998933811, 0.68671340316128552120876657741, 0.842447520547978161789360708025, 0.977529502770832894696557062749, 2.11830791649777830270543068506, 2.21423970157092315197588024821, 2.25810018513896790548440165901, 2.46253907578939283267205469826, 3.07598752514072215909753051672, 3.43063295030872095826499338234, 3.73243684450409317502394283715, 4.00333840680333740177532797361, 4.72504937983569954058748158577, 4.75400070048858678017954387518, 4.87053457832152162721405372955, 5.28528784141347941295761509725, 5.57364347261514177000946544966, 5.78943584648158753039347791396, 6.10365673643291093943405340781, 6.18874456559915966245969273110, 7.03503218981262130220260253290, 7.28985246771308818698677641825, 7.30902306289940021185878171534, 7.40037343241422406961754195505, 7.70104550776557944944898344050