L(s) = 1 | − 20·7-s + 40·13-s + 23·16-s − 40·25-s + 32·31-s + 40·37-s + 40·43-s + 200·49-s − 232·61-s + 280·67-s + 220·73-s − 800·91-s − 20·97-s − 140·103-s − 460·112-s + 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 800·169-s + 173-s + ⋯ |
L(s) = 1 | − 2.85·7-s + 3.07·13-s + 1.43·16-s − 8/5·25-s + 1.03·31-s + 1.08·37-s + 0.930·43-s + 4.08·49-s − 3.80·61-s + 4.17·67-s + 3.01·73-s − 8.79·91-s − 0.206·97-s − 1.35·103-s − 4.10·112-s + 0.132·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4.73·169-s + 0.00578·173-s + ⋯ |
Λ(s)=(=(4100625s/2ΓC(s)4L(s)Λ(3−s)
Λ(s)=(=(4100625s/2ΓC(s+1)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
4100625
= 38⋅54
|
Sign: |
1
|
Analytic conductor: |
2.26042 |
Root analytic conductor: |
1.10732 |
Motivic weight: |
2 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 4100625, ( :1,1,1,1), 1)
|
Particular Values
L(23) |
≈ |
1.048905771 |
L(21) |
≈ |
1.048905771 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 3 | | 1 |
| 5 | C22 | 1+8pT2+p4T4 |
good | 2 | C23 | 1−23T4+p8T8 |
| 7 | C22 | (1+10T+50T2+10p2T3+p4T4)2 |
| 11 | C22 | (1−8T2+p4T4)2 |
| 13 | C22 | (1−20T+200T2−20p2T3+p4T4)2 |
| 17 | C23 | 1+144322T4+p8T8 |
| 19 | C22 | (1−398T2+p4T4)2 |
| 23 | C23 | 1+517762T4+p8T8 |
| 29 | C22 | (1+568T2+p4T4)2 |
| 31 | C2 | (1−8T+p2T2)4 |
| 37 | C22 | (1−20T+200T2−20p2T3+p4T4)2 |
| 41 | C22 | (1+2362T2+p4T4)2 |
| 43 | C22 | (1−20T+200T2−20p2T3+p4T4)2 |
| 47 | C23 | 1−8681918T4+p8T8 |
| 53 | C23 | 1+3037282T4+p8T8 |
| 59 | C22 | (1−4712T2+p4T4)2 |
| 61 | C2 | (1+58T+p2T2)4 |
| 67 | C22 | (1−140T+9800T2−140p2T3+p4T4)2 |
| 71 | C22 | (1+6082T2+p4T4)2 |
| 73 | C22 | (1−110T+6050T2−110p2T3+p4T4)2 |
| 79 | C22 | (1−12338T2+p4T4)2 |
| 83 | C23 | 1−30948638T4+p8T8 |
| 89 | C1×C1 | (1−pT)4(1+pT)4 |
| 97 | C22 | (1+10T+50T2+10p2T3+p4T4)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
−11.55497479671012218091792319421, −11.25567763653390900346230388590, −10.99449609383926143649597317802, −10.70539611934926640009774025482, −10.22841955442148026022444801860, −10.09830133353748576268061770092, −9.531570061267914645515259405950, −9.460043886029300855700829233784, −9.260270254761943393254256602495, −8.715102600631252747470021527276, −8.143899248717885305180623709484, −8.107359053703639145676963085828, −7.75689356288320329873923647539, −7.01301774144573481201652635605, −6.61915701757328771252369196864, −6.34599495110069946504508343750, −6.05875941023193682378623855277, −5.85988594623789790079338906672, −5.45630548153638002365264185868, −4.47481000161295481147897288915, −3.82324094743719083889806152585, −3.46164702436307439002314720361, −3.44484010972461472724905766649, −2.52187720425739144518409042233, −1.02551862370993760166493414529,
1.02551862370993760166493414529, 2.52187720425739144518409042233, 3.44484010972461472724905766649, 3.46164702436307439002314720361, 3.82324094743719083889806152585, 4.47481000161295481147897288915, 5.45630548153638002365264185868, 5.85988594623789790079338906672, 6.05875941023193682378623855277, 6.34599495110069946504508343750, 6.61915701757328771252369196864, 7.01301774144573481201652635605, 7.75689356288320329873923647539, 8.107359053703639145676963085828, 8.143899248717885305180623709484, 8.715102600631252747470021527276, 9.260270254761943393254256602495, 9.460043886029300855700829233784, 9.531570061267914645515259405950, 10.09830133353748576268061770092, 10.22841955442148026022444801860, 10.70539611934926640009774025482, 10.99449609383926143649597317802, 11.25567763653390900346230388590, 11.55497479671012218091792319421