L(s) = 1 | + 9·9-s − 6·19-s + 14·29-s + 12·31-s − 44·41-s + 9·49-s − 22·59-s − 32·61-s − 52·79-s + 34·81-s + 12·89-s − 8·101-s − 6·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 11·169-s − 54·171-s + 173-s + 179-s + ⋯ |
L(s) = 1 | + 3·9-s − 1.37·19-s + 2.59·29-s + 2.15·31-s − 6.87·41-s + 9/7·49-s − 2.86·59-s − 4.09·61-s − 5.85·79-s + 34/9·81-s + 1.27·89-s − 0.796·101-s − 0.574·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.846·169-s − 4.12·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
Λ(s)=(=((218⋅512⋅196)s/2ΓC(s)6L(s)Λ(2−s)
Λ(s)=(=((218⋅512⋅196)s/2ΓC(s+1/2)6L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
2.847515695 |
L(21) |
≈ |
2.847515695 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
| 19 | (1+T)6 |
good | 3 | 1−p2T2+47T4−170T6+47p2T8−p6T10+p6T12 |
| 7 | 1−9T2+5pT4−38T6+5p3T8−9p4T10+p6T12 |
| 11 | (1+5T2+16T3+5pT4+p3T6)2 |
| 13 | 1+11T2+55T4−2146T6+55p2T8+11p4T10+p6T12 |
| 17 | 1−5pT2+3219T4−70126T6+3219p2T8−5p5T10+p6T12 |
| 23 | 1−97T2+4419T4−124918T6+4419p2T8−97p4T10+p6T12 |
| 29 | (1−7T+43T2−114T3+43pT4−7p2T5+p3T6)2 |
| 31 | (1−6T+77T2−340T3+77pT4−6p2T5+p3T6)2 |
| 37 | 1−2pT2+4747T4−190436T6+4747p2T8−2p5T10+p6T12 |
| 41 | (1+22T+223T2+1572T3+223pT4+22p2T5+p3T6)2 |
| 43 | 1−94T2+7591T4−340324T6+7591p2T8−94p4T10+p6T12 |
| 47 | 1−58T2+3567T4−270316T6+3567p2T8−58p4T10+p6T12 |
| 53 | 1+67T2+7527T4+328942T6+7527p2T8+67p4T10+p6T12 |
| 59 | (1+11T+37T2−246T3+37pT4+11p2T5+p3T6)2 |
| 61 | (1+16T+207T2+1600T3+207pT4+16p2T5+p3T6)2 |
| 67 | 1−3pT2+22943T4−1802666T6+22943p2T8−3p5T10+p6T12 |
| 71 | (1+pT2)6 |
| 73 | 1−69T2+14051T4−586094T6+14051p2T8−69p4T10+p6T12 |
| 79 | (1+26T+445T2+4604T3+445pT4+26p2T5+p3T6)2 |
| 83 | 1−398T2+71991T4−7610180T6+71991p2T8−398p4T10+p6T12 |
| 89 | (1−6T+15T2+188T3+15pT4−6p2T5+p3T6)2 |
| 97 | 1−402T2+79331T4−9565460T6+79331p2T8−402p4T10+p6T12 |
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L(s)=p∏ j=1∏12(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−4.49502115445240503263295553096, −4.30813300771414593835423957170, −4.09251266842098676810205321403, −4.05963809433703045354104343816, −3.84189747281471888938144246313, −3.59863614099654085249728311458, −3.55674168602339080079632327461, −3.35688173548907410371942501612, −3.28791847174755175056626734007, −3.04870458607400675070429381942, −2.89248786738415900704407369059, −2.69703934389160025444259154045, −2.50850159937470630845207701855, −2.47954697627113768009821596601, −2.46380480121972654620512450713, −1.99254294041878720120639405390, −1.64797676790851794209185882959, −1.51051979957863450251381268019, −1.46254025385839610955371610446, −1.42479544875468443842243438656, −1.38505100929604384259484377236, −1.28158527650333582157824441116, −0.67022896519519713265701968297, −0.26018709799994795815547576798, −0.25823291424071232560391514923,
0.25823291424071232560391514923, 0.26018709799994795815547576798, 0.67022896519519713265701968297, 1.28158527650333582157824441116, 1.38505100929604384259484377236, 1.42479544875468443842243438656, 1.46254025385839610955371610446, 1.51051979957863450251381268019, 1.64797676790851794209185882959, 1.99254294041878720120639405390, 2.46380480121972654620512450713, 2.47954697627113768009821596601, 2.50850159937470630845207701855, 2.69703934389160025444259154045, 2.89248786738415900704407369059, 3.04870458607400675070429381942, 3.28791847174755175056626734007, 3.35688173548907410371942501612, 3.55674168602339080079632327461, 3.59863614099654085249728311458, 3.84189747281471888938144246313, 4.05963809433703045354104343816, 4.09251266842098676810205321403, 4.30813300771414593835423957170, 4.49502115445240503263295553096
Plot not available for L-functions of degree greater than 10.