L(s) = 1 | − 11·4-s − 28·7-s + 22·13-s + 45·16-s − 132·19-s + 308·28-s − 126·31-s + 354·37-s + 272·43-s + 490·49-s − 242·52-s − 1.17e3·61-s − 363·64-s + 38·67-s + 188·73-s + 1.45e3·76-s − 690·79-s − 616·91-s − 1.98e3·97-s + 1.98e3·103-s − 2.55e3·109-s − 1.26e3·112-s − 4.75e3·121-s + 1.38e3·124-s + 127-s + 131-s + 3.69e3·133-s + ⋯ |
L(s) = 1 | − 1.37·4-s − 1.51·7-s + 0.469·13-s + 0.703·16-s − 1.59·19-s + 2.07·28-s − 0.730·31-s + 1.57·37-s + 0.964·43-s + 10/7·49-s − 0.645·52-s − 2.45·61-s − 0.708·64-s + 0.0692·67-s + 0.301·73-s + 2.19·76-s − 0.982·79-s − 0.709·91-s − 2.07·97-s + 1.89·103-s − 2.24·109-s − 1.06·112-s − 3.57·121-s + 1.00·124-s + 0.000698·127-s + 0.000666·131-s + 2.40·133-s + ⋯ |
Λ(s)=(=((38⋅58⋅74)s/2ΓC(s)4L(s)Λ(4−s)
Λ(s)=(=((38⋅58⋅74)s/2ΓC(s+3/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
38⋅58⋅74
|
Sign: |
1
|
Analytic conductor: |
7.45738×107 |
Root analytic conductor: |
9.63991 |
Motivic weight: |
3 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
4
|
Selberg data: |
(8, 38⋅58⋅74, ( :3/2,3/2,3/2,3/2), 1)
|
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 5 | | 1 |
| 7 | C1 | (1+pT)4 |
good | 2 | C22≀C2 | 1+11T2+19p2T4+11p6T6+p12T8 |
| 11 | C22≀C2 | 1+4757T2+9131212T4+4757p6T6+p12T8 |
| 13 | D4 | (1−11T+334pT2−11p3T3+p6T4)2 |
| 17 | C22≀C2 | 1+16103T2+110752648T4+16103p6T6+p12T8 |
| 19 | D4 | (1+66T+762pT2+66p3T3+p6T4)2 |
| 23 | C22≀C2 | 1+16580T2+227073517T4+16580p6T6+p12T8 |
| 29 | C22≀C2 | 1+57852T2+1698017789T4+57852p6T6+p12T8 |
| 31 | D4 | (1+63T+42068T2+63p3T3+p6T4)2 |
| 37 | D4 | (1−177T+90632T2−177p3T3+p6T4)2 |
| 41 | C22≀C2 | 1+93635T2+8952796516T4+93635p6T6+p12T8 |
| 43 | D4 | (1−136T+147517T2−136p3T3+p6T4)2 |
| 47 | C22≀C2 | 1+174744T2+15146248718T4+174744p6T6+p12T8 |
| 53 | C22≀C2 | 1+494267T2+102952583068T4+494267p6T6+p12T8 |
| 59 | C22≀C2 | 1+646299T2+187556725712T4+646299p6T6+p12T8 |
| 61 | D4 | (1+585T+372962T2+585p3T3+p6T4)2 |
| 67 | D4 | (1−19T+404134T2−19p3T3+p6T4)2 |
| 71 | C22≀C2 | 1+397485T2+173980338356T4+397485p6T6+p12T8 |
| 73 | D4 | (1−94T+606202T2−94p3T3+p6T4)2 |
| 79 | D4 | (1+345T+1001934T2+345p3T3+p6T4)2 |
| 83 | C22≀C2 | 1+1768119T2+1429880388068T4+1768119p6T6+p12T8 |
| 89 | C22≀C2 | 1+1398400T2+1454863323486T4+1398400p6T6+p12T8 |
| 97 | D4 | (1+990T+1951602T2+990p3T3+p6T4)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
−6.63696174865574883611972844360, −6.52948058013293642215949434825, −6.20642830598497974048347734189, −6.17110395030828208201658990292, −6.10702880066989189109128303553, −5.60035410689743955010456760449, −5.47363517113385842461577051025, −5.26852909626236097040674487360, −5.14334382571885853409104576655, −4.59197763588451983805407658445, −4.46219546241992902138788079343, −4.33825213797983399165174840191, −4.25495826248723634826451929876, −3.77225699248095646594731869118, −3.75373105645356305031978132027, −3.52074950236082126130273708813, −3.14588794821273491207127706704, −2.79404003475741885108642862281, −2.76650522359898395153222623748, −2.35800383015989326197550826542, −2.17412756180701093862960969476, −1.78746356226343367574356846662, −1.18601180477300205568132783563, −1.07983027148695969248977662005, −1.01983443144480862861872200845, 0, 0, 0, 0,
1.01983443144480862861872200845, 1.07983027148695969248977662005, 1.18601180477300205568132783563, 1.78746356226343367574356846662, 2.17412756180701093862960969476, 2.35800383015989326197550826542, 2.76650522359898395153222623748, 2.79404003475741885108642862281, 3.14588794821273491207127706704, 3.52074950236082126130273708813, 3.75373105645356305031978132027, 3.77225699248095646594731869118, 4.25495826248723634826451929876, 4.33825213797983399165174840191, 4.46219546241992902138788079343, 4.59197763588451983805407658445, 5.14334382571885853409104576655, 5.26852909626236097040674487360, 5.47363517113385842461577051025, 5.60035410689743955010456760449, 6.10702880066989189109128303553, 6.17110395030828208201658990292, 6.20642830598497974048347734189, 6.52948058013293642215949434825, 6.63696174865574883611972844360