L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s + 3·9-s + 4·15-s − 8·17-s − 8·19-s − 4·21-s + 10·23-s + 25-s + 10·27-s − 6·29-s − 12·31-s − 4·35-s − 24·37-s + 10·41-s − 4·43-s + 6·45-s + 14·47-s + 9·49-s − 16·51-s − 8·53-s − 16·57-s + 12·59-s + 6·61-s − 6·63-s + 2·67-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s + 1.03·15-s − 1.94·17-s − 1.83·19-s − 0.872·21-s + 2.08·23-s + 1/5·25-s + 1.92·27-s − 1.11·29-s − 2.15·31-s − 0.676·35-s − 3.94·37-s + 1.56·41-s − 0.609·43-s + 0.894·45-s + 2.04·47-s + 9/7·49-s − 2.24·51-s − 1.09·53-s − 2.11·57-s + 1.56·59-s + 0.768·61-s − 0.755·63-s + 0.244·67-s + ⋯ |
Λ(s)=(=((212⋅38⋅54)s/2ΓC(s)4L(s)Λ(2−s)
Λ(s)=(=((212⋅38⋅54)s/2ΓC(s+1/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
212⋅38⋅54
|
Sign: |
1
|
Analytic conductor: |
68.2839 |
Root analytic conductor: |
1.69546 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 212⋅38⋅54, ( :1/2,1/2,1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
2.483836693 |
L(21) |
≈ |
2.483836693 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | | 1 |
| 3 | C22 | 1−2T+T2−2pT3+p2T4 |
| 5 | C2 | (1−T+T2)2 |
good | 7 | D4×C2 | 1+2T−5T2−10T3+4T4−10pT5−5p2T6+2p3T7+p4T8 |
| 11 | C22 | (1−pT2+p2T4)2 |
| 13 | C22 | (1−pT2+p2T4)2 |
| 17 | C2 | (1+2T+pT2)4 |
| 19 | D4 | (1+4T+18T2+4pT3+p2T4)2 |
| 23 | D4×C2 | 1−10T+35T2−190T3+1396T4−190pT5+35p2T6−10p3T7+p4T8 |
| 29 | D4×C2 | 1+6T−7T2−90T3−36T4−90pT5−7p2T6+6p3T7+p4T8 |
| 31 | D4×C2 | 1+12T+70T2+144T3+51T4+144pT5+70p2T6+12p3T7+p4T8 |
| 37 | C2 | (1+6T+pT2)4 |
| 41 | D4×C2 | 1−10T+17T2−10T3+1108T4−10pT5+17p2T6−10p3T7+p4T8 |
| 43 | D4×C2 | 1+4T+22T2−368T3−2501T4−368pT5+22p2T6+4p3T7+p4T8 |
| 47 | D4×C2 | 1−14T+59T2−602T3+7348T4−602pT5+59p2T6−14p3T7+p4T8 |
| 53 | D4 | (1+4T+14T2+4pT3+p2T4)2 |
| 59 | D4×C2 | 1−12T+14T2−144T3+4923T4−144pT5+14p2T6−12p3T7+p4T8 |
| 61 | C22 | (1−3T−52T2−3pT3+p2T4)2 |
| 67 | D4×C2 | 1−2T+19T2+298T3−4532T4+298pT5+19p2T6−2p3T7+p4T8 |
| 71 | C22 | (1+46T2+p2T4)2 |
| 73 | D4 | (1−8T+66T2−8pT3+p2T4)2 |
| 79 | D4×C2 | 1−4T−122T2+80T3+11539T4+80pT5−122p2T6−4p3T7+p4T8 |
| 83 | D4×C2 | 1−6T−133T2−18T3+18684T4−18pT5−133p2T6−6p3T7+p4T8 |
| 89 | D4 | (1+14T+131T2+14pT3+p2T4)2 |
| 97 | C22 | (1+2T−93T2+2pT3+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
−8.410216165895431156306392478615, −8.213947696406722007585818302209, −7.81992017963868845293862917471, −7.58033622860044370313005924892, −7.07107515254452554170514653658, −6.92719753273848474696197163068, −6.80474761404584793533608593350, −6.73509164905450659471738177886, −6.56996510580662521243563312487, −5.99258756844737649999135968223, −5.56119023600477559060575936366, −5.46436336555606318863860802678, −5.21943140144790973909746308195, −5.03210393698369414987593930865, −4.31804690430714939877816087568, −4.21677105280676442361984230932, −3.99071748656861140630327823532, −3.70179882711711695871574156615, −3.18251269739172063177625062524, −2.98189148802267898869008997397, −2.50688123047966187989500561819, −2.26102702955856286519722017493, −1.80144596751132884212957624817, −1.69963129427529424521091621478, −0.55773988127577540184293511907,
0.55773988127577540184293511907, 1.69963129427529424521091621478, 1.80144596751132884212957624817, 2.26102702955856286519722017493, 2.50688123047966187989500561819, 2.98189148802267898869008997397, 3.18251269739172063177625062524, 3.70179882711711695871574156615, 3.99071748656861140630327823532, 4.21677105280676442361984230932, 4.31804690430714939877816087568, 5.03210393698369414987593930865, 5.21943140144790973909746308195, 5.46436336555606318863860802678, 5.56119023600477559060575936366, 5.99258756844737649999135968223, 6.56996510580662521243563312487, 6.73509164905450659471738177886, 6.80474761404584793533608593350, 6.92719753273848474696197163068, 7.07107515254452554170514653658, 7.58033622860044370313005924892, 7.81992017963868845293862917471, 8.213947696406722007585818302209, 8.410216165895431156306392478615