L(s) = 1 | + 4-s + 4·5-s + 9-s + 10·11-s − 14·19-s + 4·20-s + 5·25-s − 12·31-s + 36-s + 36·41-s + 10·44-s + 4·45-s + 40·55-s − 8·59-s − 4·61-s − 64-s − 8·71-s − 14·76-s − 28·79-s − 20·89-s − 56·95-s + 10·99-s + 5·100-s + 16·101-s − 36·109-s + 47·121-s − 12·124-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 1.78·5-s + 1/3·9-s + 3.01·11-s − 3.21·19-s + 0.894·20-s + 25-s − 2.15·31-s + 1/6·36-s + 5.62·41-s + 1.50·44-s + 0.596·45-s + 5.39·55-s − 1.04·59-s − 0.512·61-s − 1/8·64-s − 0.949·71-s − 1.60·76-s − 3.15·79-s − 2.11·89-s − 5.74·95-s + 1.00·99-s + 1/2·100-s + 1.59·101-s − 3.44·109-s + 4.27·121-s − 1.07·124-s + ⋯ |
Λ(s)=(=((24⋅34⋅54⋅78)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((24⋅34⋅54⋅78)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
24⋅34⋅54⋅78
|
Sign: |
1
|
Analytic conductor: |
18983.5 |
Root analytic conductor: |
3.42607 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 24⋅34⋅54⋅78, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
4.836592647 |
L(21) |
≈ |
4.836592647 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C22 | 1−T2+T4 |
| 3 | C22 | 1−T2+T4 |
| 5 | C22 | 1−4T+11T2−4pT3+p2T4 |
| 7 | | 1 |
good | 11 | C22 | (1−5T+14T2−5pT3+p2T4)2 |
| 13 | C22 | (1−25T2+p2T4)2 |
| 17 | C22×C22 | (1−8T+47T2−8pT3+p2T4)(1+8T+47T2+8pT3+p2T4) |
| 19 | C2 | (1−T+pT2)2(1+8T+pT2)2 |
| 23 | C23 | 1+37T2+840T4+37p2T6+p4T8 |
| 29 | C2 | (1+pT2)4 |
| 31 | C22 | (1+6T+5T2+6pT3+p2T4)2 |
| 37 | C23 | 1+49T2+1032T4+49p2T6+p4T8 |
| 41 | C2 | (1−9T+pT2)4 |
| 43 | C22 | (1+14T2+p2T4)2 |
| 47 | C23 | 1−75T2+3416T4−75p2T6+p4T8 |
| 53 | C23 | 1+105T2+8216T4+105p2T6+p4T8 |
| 59 | C22 | (1+4T−43T2+4pT3+p2T4)2 |
| 61 | C22 | (1+2T−57T2+2pT3+p2T4)2 |
| 67 | C23 | 1+98T2+5115T4+98p2T6+p4T8 |
| 71 | C2 | (1+2T+pT2)4 |
| 73 | C23 | 1+130T2+11571T4+130p2T6+p4T8 |
| 79 | C22 | (1+14T+117T2+14pT3+p2T4)2 |
| 83 | C22 | (1−66T2+p2T4)2 |
| 89 | C22 | (1+10T+11T2+10pT3+p2T4)2 |
| 97 | C2 | (1−18T+pT2)2(1+18T+pT2)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
−6.70961437116929632648492208728, −6.35626794314856406408041272069, −6.29448405768742356315335945177, −6.24123521958903892584338073862, −6.17206546255510874341110078811, −5.62939234406027827604453509770, −5.57220493575768833728711896508, −5.46874296862761588396016630138, −5.25479737088151368307019615635, −4.48580288724954478520821387002, −4.31376024040273172655938024966, −4.25250627929462847079527339121, −4.25231144002982700438526931217, −4.06546125420022607425354934876, −3.73455847310880372740679756228, −3.17778809265516785632724721791, −3.07274155075159331636336727263, −2.64420450594449491502242355548, −2.31797635134604043994652416418, −2.22934482189743452760916840784, −1.90965459849557283694028087086, −1.55372784150797293839212904482, −1.25883331463203513692296584288, −1.25792023263463398658579996005, −0.35103269051662378624537017799,
0.35103269051662378624537017799, 1.25792023263463398658579996005, 1.25883331463203513692296584288, 1.55372784150797293839212904482, 1.90965459849557283694028087086, 2.22934482189743452760916840784, 2.31797635134604043994652416418, 2.64420450594449491502242355548, 3.07274155075159331636336727263, 3.17778809265516785632724721791, 3.73455847310880372740679756228, 4.06546125420022607425354934876, 4.25231144002982700438526931217, 4.25250627929462847079527339121, 4.31376024040273172655938024966, 4.48580288724954478520821387002, 5.25479737088151368307019615635, 5.46874296862761588396016630138, 5.57220493575768833728711896508, 5.62939234406027827604453509770, 6.17206546255510874341110078811, 6.24123521958903892584338073862, 6.29448405768742356315335945177, 6.35626794314856406408041272069, 6.70961437116929632648492208728