L(s) = 1 | − 6·2-s − 6·3-s + 23·4-s − 4·5-s + 36·6-s − 7·7-s − 72·8-s + 13·9-s + 24·10-s + 2·11-s − 138·12-s − 6·13-s + 42·14-s + 24·15-s + 199·16-s + 8·17-s − 78·18-s − 5·19-s − 92·20-s + 42·21-s − 12·22-s + 32·23-s + 432·24-s + 26·25-s + 36·26-s + 2·27-s − 161·28-s + ⋯ |
L(s) = 1 | − 4.24·2-s − 3.46·3-s + 23/2·4-s − 1.78·5-s + 14.6·6-s − 2.64·7-s − 25.4·8-s + 13/3·9-s + 7.58·10-s + 0.603·11-s − 39.8·12-s − 1.66·13-s + 11.2·14-s + 6.19·15-s + 49.7·16-s + 1.94·17-s − 18.3·18-s − 1.14·19-s − 20.5·20-s + 9.16·21-s − 2.55·22-s + 6.67·23-s + 88.1·24-s + 26/5·25-s + 7.06·26-s + 0.384·27-s − 30.4·28-s + ⋯ |
Λ(s)=(=((3116)s/2ΓC(s)8L(s)Λ(2−s)
Λ(s)=(=((3116)s/2ΓC(s+1/2)8L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
0.01723500717 |
L(21) |
≈ |
0.01723500717 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 31 | 1 |
good | 2 | (1+3T+pT2+T4+p3T6+3p3T7+p4T8)2 |
| 3 | 1+2pT+23T2+58T3+4p3T4+148T5+47pT6+94T7+43T8+94pT9+47p3T10+148p3T11+4p7T12+58p5T13+23p6T14+2p8T15+p8T16 |
| 5 | (1+T−4T2+pT3+p2T4)4 |
| 7 | 1+pT+37T2+134T3+458T4+1471T5+4694T6+14192T7+39103T8+14192pT9+4694p2T10+1471p3T11+458p4T12+134p5T13+37p6T14+p8T15+p8T16 |
| 11 | 1−2T+pT2−58T3+116T4+164T5+39pT6+4756T7−24153T8+4756pT9+39p3T10+164p3T11+116p4T12−58p5T13+p7T14−2p7T15+p8T16 |
| 13 | 1+6T+33T2+38T3−12T4−1132T5+127pT6+16194T7+133363T8+16194pT9+127p3T10−1132p3T11−12p4T12+38p5T13+33p6T14+6p7T15+p8T16 |
| 17 | 1−8T+57T2−276T3+1508T4−6624T5+35159T6−145358T7+660723T8−145358pT9+35159p2T10−6624p3T11+1508p4T12−276p5T13+57p6T14−8p7T15+p8T16 |
| 19 | 1+5T+29T2−70T3−590T4−4435T5−2714T6+42900T7+426339T8+42900pT9−2714p2T10−4435p3T11−590p4T12−70p5T13+29p6T14+5p7T15+p8T16 |
| 23 | (1−16T+113T2−570T3+2741T4−570pT5+113p2T6−16p3T7+p4T8)2 |
| 29 | (1+11T2−90T3+661T4−90pT5+11p2T6+p4T8)2 |
| 37 | (1−2T−33T2−2pT3+p2T4)4 |
| 41 | 1−7T+pT2−518T3+3626T4−15911T5+3954pT6−1177414T7+5416137T8−1177414pT9+3954p3T10−15911p3T11+3626p4T12−518p5T13+p7T14−7p7T15+p8T16 |
| 43 | 1−4T+pT2+328T3−2092T4+17288T5−42379T6−635666T7+2926363T8−635666pT9−42379p2T10+17288p3T11−2092p4T12+328p5T13+p7T14−4p7T15+p8T16 |
| 47 | (1+8T+17T2+380T3+4721T4+380pT5+17p2T6+8p3T7+p4T8)2 |
| 53 | 1−4T−27T2+588T3−4432T4+1608T5+100691T6−1155796T7+6693663T8−1155796pT9+100691p2T10+1608p3T11−4432p4T12+588p5T13−27p6T14−4p7T15+p8T16 |
| 59 | 1−5T+69T2+270T3−1990T4+37635T5−145834T6−485900T7+5823099T8−485900pT9−145834p2T10+37635p3T11−1990p4T12+270p5T13+69p6T14−5p7T15+p8T16 |
| 61 | (1+6T+6T2+6pT3+p2T4)4 |
| 67 | (1+8T−3T2+8pT3+p2T4)4 |
| 71 | 1−27T+421T2−3318T3+6786T4+165549T5−1786666T6+6219816T7−615913T8+6219816pT9−1786666p2T10+165549p3T11+6786p4T12−3318p5T13+421p6T14−27p7T15+p8T16 |
| 73 | 1+6T+33T2−1262T3−12972T4−61052T5+133831T6+7999944T7+53220103T8+7999944pT9+133831p2T10−61052p3T11−12972p4T12−1262p5T13+33p6T14+6p7T15+p8T16 |
| 79 | 1−10T+19T2+1570T3−21460T4+132020T5+181241T6−13939970T7+166725259T8−13939970pT9+181241p2T10+132020p3T11−21460p4T12+1570p5T13+19p6T14−10p7T15+p8T16 |
| 83 | 1+26T+483T2+5778T3+60548T4+592908T5+6213461T6+66709124T7+637073703T8+66709124pT9+6213461p2T10+592908p3T11+60548p4T12+5778p5T13+483p6T14+26p7T15+p8T16 |
| 89 | (1−10T+71T2−1290T3+19001T4−1290pT5+71p2T6−10p3T7+p4T8)2 |
| 97 | (1+13T+192T2+2195T3+31031T4+2195pT5+192p2T6+13p3T7+p4T8)2 |
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L(s)=p∏ j=1∏16(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−4.49684487333086695161552711886, −4.49483721096350252278023941831, −3.63527944416769712697640294980, −3.58996063837341351886563394607, −3.49471874149431193227568583481, −3.44524718485351057974136785103, −3.39701402242738453076934934571, −3.30716302845353359267805451433, −3.27615575302814060441686210196, −3.02483050540509451531363459001, −2.69134623497512939176059267328, −2.61222208103411756529330113997, −2.58292996710064620528850832071, −2.58152695877098653976313965568, −2.42978022294154093150363102381, −2.41733248453880606556833348629, −1.43357197271161183365410659094, −1.26205335607265657257296920292, −1.26182388216994766718607037727, −1.18765799705811748491905021439, −1.17773583151605095628262952506, −0.938791577748437918209742390881, −0.35131409606111784251594584190, −0.33573783907737316012342669251, −0.23567692889004931292023260099,
0.23567692889004931292023260099, 0.33573783907737316012342669251, 0.35131409606111784251594584190, 0.938791577748437918209742390881, 1.17773583151605095628262952506, 1.18765799705811748491905021439, 1.26182388216994766718607037727, 1.26205335607265657257296920292, 1.43357197271161183365410659094, 2.41733248453880606556833348629, 2.42978022294154093150363102381, 2.58152695877098653976313965568, 2.58292996710064620528850832071, 2.61222208103411756529330113997, 2.69134623497512939176059267328, 3.02483050540509451531363459001, 3.27615575302814060441686210196, 3.30716302845353359267805451433, 3.39701402242738453076934934571, 3.44524718485351057974136785103, 3.49471874149431193227568583481, 3.58996063837341351886563394607, 3.63527944416769712697640294980, 4.49483721096350252278023941831, 4.49684487333086695161552711886
Plot not available for L-functions of degree greater than 10.