L(s) = 1 | − 3·4-s − 4·5-s − 3·9-s + 16·11-s + 6·16-s + 4·19-s + 12·20-s + 4·25-s + 4·29-s − 6·31-s + 9·36-s + 12·41-s − 48·44-s + 12·45-s + 30·49-s − 64·55-s + 4·59-s + 28·61-s − 10·64-s − 16·71-s − 12·76-s + 12·79-s − 24·80-s + 6·81-s + 4·89-s − 16·95-s − 48·99-s + ⋯ |
L(s) = 1 | − 3/2·4-s − 1.78·5-s − 9-s + 4.82·11-s + 3/2·16-s + 0.917·19-s + 2.68·20-s + 4/5·25-s + 0.742·29-s − 1.07·31-s + 3/2·36-s + 1.87·41-s − 7.23·44-s + 1.78·45-s + 30/7·49-s − 8.62·55-s + 0.520·59-s + 3.58·61-s − 5/4·64-s − 1.89·71-s − 1.37·76-s + 1.35·79-s − 2.68·80-s + 2/3·81-s + 0.423·89-s − 1.64·95-s − 4.82·99-s + ⋯ |
Λ(s)=(=((26⋅36⋅56⋅316)s/2ΓC(s)6L(s)Λ(2−s)
Λ(s)=(=((26⋅36⋅56⋅316)s/2ΓC(s+1/2)6L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
3.992161861 |
L(21) |
≈ |
3.992161861 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | (1+T2)3 |
| 3 | (1+T2)3 |
| 5 | 1+4T+12T2+36T3+12pT4+4p2T5+p3T6 |
| 31 | (1+T)6 |
good | 7 | (1−10T2+p2T4)3 |
| 11 | (1−8T+46T2−168T3+46pT4−8p2T5+p3T6)2 |
| 13 | 1−28T2+280T4−2014T6+280p2T8−28p4T10+p6T12 |
| 17 | 1−28T2+512T4−11034T6+512p2T8−28p4T10+p6T12 |
| 19 | (1−2T+22T2−80T3+22pT4−2p2T5+p3T6)2 |
| 23 | 1−70T2+2639T4−71412T6+2639p2T8−70p4T10+p6T12 |
| 29 | (1−2T+55T2−132T3+55pT4−2p2T5+p3T6)2 |
| 37 | 1−58T2+1895T4−66348T6+1895p2T8−58p4T10+p6T12 |
| 41 | (1−2T+pT2)6 |
| 43 | 1−106T2+7367T4−400332T6+7367p2T8−106p4T10+p6T12 |
| 47 | 1−208T2+20432T4−1209294T6+20432p2T8−208p4T10+p6T12 |
| 53 | 1−130T2+4055T4+2820T6+4055p2T8−130p4T10+p6T12 |
| 59 | (1−2T+101T2+44T3+101pT4−2p2T5+p3T6)2 |
| 61 | (1−14T+92T2−336T3+92pT4−14p2T5+p3T6)2 |
| 67 | 1−116T2+11776T4−712958T6+11776p2T8−116p4T10+p6T12 |
| 71 | (1+8T+134T2+1156T3+134pT4+8p2T5+p3T6)2 |
| 73 | 1−170T2+20671T4−1830476T6+20671p2T8−170p4T10+p6T12 |
| 79 | (1−6T+230T2−920T3+230pT4−6p2T5+p3T6)2 |
| 83 | 1−128T2+13008T4−1479782T6+13008p2T8−128p4T10+p6T12 |
| 89 | (1−2T+123T2−36T3+123pT4−2p2T5+p3T6)2 |
| 97 | 1−432T2+89120T4−10929554T6+89120p2T8−432p4T10+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
−5.28579331807154264471005186648, −5.17898751134358155285796112113, −5.05774721293566686167443644511, −4.82320216324444104996066113937, −4.50416628739632516664040818881, −4.39630658267190707424944343763, −4.15923503549313541264775914404, −4.02776596307716053345255355874, −4.01845292749549910138937693410, −3.77988037861326749413454058666, −3.76122966367400274574582673828, −3.72840688322061401172338255051, −3.66575857188705674443097216072, −2.99862139772311652847097024544, −2.94188179586906768208890435097, −2.82763970041976481777680247682, −2.73154872617466493573894720637, −2.09185019993051032073452075394, −2.04695849668419524796122432955, −1.63117750483127575910222366453, −1.57738892623066139293423674829, −1.01461484730051513890731435765, −0.78776609613215060268633625533, −0.75028415678710255794022844439, −0.53985296122424655334859698180,
0.53985296122424655334859698180, 0.75028415678710255794022844439, 0.78776609613215060268633625533, 1.01461484730051513890731435765, 1.57738892623066139293423674829, 1.63117750483127575910222366453, 2.04695849668419524796122432955, 2.09185019993051032073452075394, 2.73154872617466493573894720637, 2.82763970041976481777680247682, 2.94188179586906768208890435097, 2.99862139772311652847097024544, 3.66575857188705674443097216072, 3.72840688322061401172338255051, 3.76122966367400274574582673828, 3.77988037861326749413454058666, 4.01845292749549910138937693410, 4.02776596307716053345255355874, 4.15923503549313541264775914404, 4.39630658267190707424944343763, 4.50416628739632516664040818881, 4.82320216324444104996066113937, 5.05774721293566686167443644511, 5.17898751134358155285796112113, 5.28579331807154264471005186648
Plot not available for L-functions of degree greater than 10.