L(s) = 1 | − 36·3-s + 810·9-s + 312·13-s + 384·17-s − 2.42e3·23-s + 6.62e3·25-s − 1.45e4·27-s − 3.96e3·29-s − 1.12e4·39-s − 4.83e3·43-s + 4.91e4·49-s − 1.38e4·51-s − 1.82e3·53-s − 1.00e5·61-s + 8.72e4·69-s − 2.38e5·75-s + 2.05e5·79-s + 2.29e5·81-s + 1.42e5·87-s + 1.09e5·101-s + 8.31e4·103-s − 5.00e5·107-s − 3.44e5·113-s + 2.52e5·117-s + 6.17e5·121-s + 127-s + 1.73e5·129-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 10/3·9-s + 0.512·13-s + 0.322·17-s − 0.955·23-s + 2.11·25-s − 3.84·27-s − 0.874·29-s − 1.18·39-s − 0.398·43-s + 2.92·49-s − 0.744·51-s − 0.0891·53-s − 3.47·61-s + 2.20·69-s − 4.89·75-s + 3.70·79-s + 35/9·81-s + 2.01·87-s + 1.06·101-s + 0.772·103-s − 4.22·107-s − 2.53·113-s + 1.70·117-s + 3.83·121-s + 5.50e−6·127-s + 0.920·129-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: |
0
|
Selberg data: |
(8, 216⋅34⋅134, ( :5/2,5/2,5/2,5/2), 1)
|
Particular Values
L(3) |
≈ |
0.4823859960 |
L(21) |
≈ |
0.4823859960 |
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 | D4 | 1−24pT−146p3T2−24p6T3+p10T4 |
good | 5 | C22≀C2 | 1−6624T2+1083838p2T4−6624p10T6+p20T8 |
| 7 | C22≀C2 | 1−49132T2+1140403638T4−49132p10T6+p20T8 |
| 11 | C22≀C2 | 1−617088T2+146890430542T4−617088p10T6+p20T8 |
| 17 | D4 | (1−192T+2791006T2−192p5T3+p10T4)2 |
| 19 | C22≀C2 | 1−4292764T2+9023685324870T4−4292764p10T6+p20T8 |
| 23 | D4 | (1+1212T+3450766T2+1212p5T3+p10T4)2 |
| 29 | D4 | (1+1980T+13967182T2+1980p5T3+p10T4)2 |
| 31 | C22≀C2 | 1−72558796T2+2754768158742p2T4−72558796p10T6+p20T8 |
| 37 | C22≀C2 | 1−191019364T2+17466588618543606T4−191019364p10T6+p20T8 |
| 41 | C22≀C2 | 1−259347072T2+41835734180463454T4−259347072p10T6+p20T8 |
| 43 | D4 | (1+2416T+284123046T2+2416p5T3+p10T4)2 |
| 47 | C22≀C2 | 1−407420832T2+124761975513464638T4−407420832p10T6+p20T8 |
| 53 | D4 | (1+912T+836540998T2+912p5T3+p10T4)2 |
| 59 | C22≀C2 | 1−151308144T2+442844399562129262T4−151308144p10T6+p20T8 |
| 61 | D4 | (1+50496T+2324330710T2+50496p5T3+p10T4)2 |
| 67 | C22≀C2 | 1−45114940T2−516263867957300538T4−45114940p10T6+p20T8 |
| 71 | C22≀C2 | 1−6784475376T2+18015098428504544062T4−6784475376p10T6+p20T8 |
| 73 | C22≀C2 | 1+4119671084T2+12172896815453134278T4+4119671084p10T6+p20T8 |
| 79 | D4 | (1−102672T+7610525470T2−102672p5T3+p10T4)2 |
| 83 | C22≀C2 | 1−12846419184T2+72288762441539676238T4−12846419184p10T6+p20T8 |
| 89 | C22≀C2 | 1+9461922432T2+70529599078905653662T4+9461922432p10T6+p20T8 |
| 97 | C22≀C2 | 1−23763592180T2+28⋯94T4−23763592180p10T6+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
−6.73548468769692341503113403145, −6.58480493649862435851448345709, −6.29614761626982955008685865880, −6.06465064269600518032788290438, −5.81543646847883110422149216338, −5.71133189284422302858144532982, −5.53010982580691734201868019258, −5.14791063465802708158219883836, −4.89284365950255543310569194522, −4.70802418925461832163991844027, −4.58801467785745897756071883492, −4.13507870525019502984986974555, −4.02465113623678247740999029633, −3.58121956570458921271735987118, −3.43012528962717217230170215135, −3.12873385530515772411013333359, −2.55035684222680872200842907481, −2.48837013734943451908246258424, −1.92657821483756308793116593908, −1.65930055932560466870314747301, −1.42824408489393974091797167423, −0.987217426542848906393094559167, −0.70759572161363514706061269632, −0.62804971296690877803214317371, −0.11036701592707312095807058524,
0.11036701592707312095807058524, 0.62804971296690877803214317371, 0.70759572161363514706061269632, 0.987217426542848906393094559167, 1.42824408489393974091797167423, 1.65930055932560466870314747301, 1.92657821483756308793116593908, 2.48837013734943451908246258424, 2.55035684222680872200842907481, 3.12873385530515772411013333359, 3.43012528962717217230170215135, 3.58121956570458921271735987118, 4.02465113623678247740999029633, 4.13507870525019502984986974555, 4.58801467785745897756071883492, 4.70802418925461832163991844027, 4.89284365950255543310569194522, 5.14791063465802708158219883836, 5.53010982580691734201868019258, 5.71133189284422302858144532982, 5.81543646847883110422149216338, 6.06465064269600518032788290438, 6.29614761626982955008685865880, 6.58480493649862435851448345709, 6.73548468769692341503113403145