L(s) = 1 | + 4·3-s − 2·9-s − 12·23-s − 4·25-s − 40·27-s − 12·29-s − 12·41-s − 36·47-s + 28·49-s − 48·69-s − 16·75-s − 55·81-s − 48·87-s + 48·101-s + 4·121-s − 48·123-s + 127-s + 131-s + 137-s + 139-s − 144·141-s + 112·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 2/3·9-s − 2.50·23-s − 4/5·25-s − 7.69·27-s − 2.22·29-s − 1.87·41-s − 5.25·47-s + 4·49-s − 5.77·69-s − 1.84·75-s − 6.11·81-s − 5.14·87-s + 4.77·101-s + 4/11·121-s − 4.32·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 12.1·141-s + 9.23·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
Λ(s)=(=((216⋅54⋅234)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((216⋅54⋅234)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
216⋅54⋅234
|
Sign: |
1
|
Analytic conductor: |
46599.3 |
Root analytic conductor: |
3.83307 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 216⋅54⋅234, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
0.3985007068 |
L(21) |
≈ |
0.3985007068 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 5 | C22 | 1+4T2+p2T4 |
| 23 | C2 | (1+6T+pT2)2 |
good | 3 | C2 | (1−T+pT2)4 |
| 7 | C2 | (1−pT2)4 |
| 11 | C22 | (1−2T2+p2T4)2 |
| 13 | C22 | (1−5T2+p2T4)2 |
| 17 | C22 | (1+28T2+p2T4)2 |
| 19 | C22 | (1+32T2+p2T4)2 |
| 29 | C2 | (1+3T+pT2)4 |
| 31 | C22 | (1−41T2+p2T4)2 |
| 37 | C22 | (1+50T2+p2T4)2 |
| 41 | C2 | (1+3T+pT2)4 |
| 43 | C22 | (1+40T2+p2T4)2 |
| 47 | C2 | (1+9T+pT2)4 |
| 53 | C22 | (1+82T2+p2T4)2 |
| 59 | C22 | (1−34T2+p2T4)2 |
| 61 | C22 | (1+4T2+p2T4)2 |
| 67 | C2 | (1−pT2)4 |
| 71 | C22 | (1+47T2+p2T4)2 |
| 73 | C22 | (1−125T2+p2T4)2 |
| 79 | C22 | (1+104T2+p2T4)2 |
| 83 | C22 | (1−152T2+p2T4)2 |
| 89 | C22 | (1−164T2+p2T4)2 |
| 97 | C22 | (1+98T2+p2T4)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
−6.43478046078107329586528432807, −6.36593492147629343623786684738, −6.32191362593348365794404325148, −5.71120169066776123779638387291, −5.65384610217344348169859750497, −5.61847347114298276627142261224, −5.61274604279707611840117975904, −4.98279313822080111030086588460, −4.87528081723599115190879580249, −4.75502408066513450002033873557, −4.15111639098684444103440491684, −3.92790280088325373947609346548, −3.71371550653991960959025391469, −3.68589689846598288322817894372, −3.38263240706867473309418610355, −3.33938658896238229278194786875, −2.85686275816703161421351086564, −2.72366215782119817803986450294, −2.50161683764124524329507681317, −2.02074777555634067416883342830, −2.00231717390234486346829632302, −1.83860759414197422465421681256, −1.54951259656422904729164029608, −0.54267309294496647913256063402, −0.11866920241489012450222362196,
0.11866920241489012450222362196, 0.54267309294496647913256063402, 1.54951259656422904729164029608, 1.83860759414197422465421681256, 2.00231717390234486346829632302, 2.02074777555634067416883342830, 2.50161683764124524329507681317, 2.72366215782119817803986450294, 2.85686275816703161421351086564, 3.33938658896238229278194786875, 3.38263240706867473309418610355, 3.68589689846598288322817894372, 3.71371550653991960959025391469, 3.92790280088325373947609346548, 4.15111639098684444103440491684, 4.75502408066513450002033873557, 4.87528081723599115190879580249, 4.98279313822080111030086588460, 5.61274604279707611840117975904, 5.61847347114298276627142261224, 5.65384610217344348169859750497, 5.71120169066776123779638387291, 6.32191362593348365794404325148, 6.36593492147629343623786684738, 6.43478046078107329586528432807