L(s) = 1 | − 3·2-s + 3·3-s + 3·4-s + 2·5-s − 9·6-s + 12·7-s + 2·8-s + 3·9-s − 6·10-s + 6·11-s + 9·12-s + 3·13-s − 36·14-s + 6·15-s − 9·16-s − 2·17-s − 9·18-s + 3·19-s + 6·20-s + 36·21-s − 18·22-s + 10·23-s + 6·24-s + 9·25-s − 9·26-s − 2·27-s + 36·28-s + ⋯ |
L(s) = 1 | − 2.12·2-s + 1.73·3-s + 3/2·4-s + 0.894·5-s − 3.67·6-s + 4.53·7-s + 0.707·8-s + 9-s − 1.89·10-s + 1.80·11-s + 2.59·12-s + 0.832·13-s − 9.62·14-s + 1.54·15-s − 9/4·16-s − 0.485·17-s − 2.12·18-s + 0.688·19-s + 1.34·20-s + 7.85·21-s − 3.83·22-s + 2.08·23-s + 1.22·24-s + 9/5·25-s − 1.76·26-s − 0.384·27-s + 6.80·28-s + ⋯ |
Λ(s)=(=((26⋅36⋅116⋅196)s/2ΓC(s)6L(s)Λ(2−s)
Λ(s)=(=((26⋅36⋅116⋅196)s/2ΓC(s+1/2)6L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
4.895290416 |
L(21) |
≈ |
4.895290416 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | (1+T+T2)3 |
| 3 | (1−T+T2)3 |
| 11 | (1−T)6 |
| 19 | 1−3T−6T2+43T3−6pT4−3p2T5+p3T6 |
good | 5 | 1−2T−pT2+6T3+12T4+22T5−111T6+22pT7+12p2T8+6p3T9−p5T10−2p5T11+p6T12 |
| 7 | (1−6T+26T2−76T3+26pT4−6p2T5+p3T6)2 |
| 13 | 1−3T+6T2+3pT3−102T4−345T5+3332T6−345pT7−102p2T8+3p4T9+6p4T10−3p5T11+p6T12 |
| 17 | 1+2T−16T2−184T3−164T4+1458T5+14414T6+1458pT7−164p2T8−184p3T9−16p4T10+2p5T11+p6T12 |
| 23 | 1−10T+14T2+68T3+778T4−5634T5+16622T6−5634pT7+778p2T8+68p3T9+14p4T10−10p5T11+p6T12 |
| 29 | 1−3T−47T2−38T3+1317T4+3581T5−45510T6+3581pT7+1317p2T8−38p3T9−47p4T10−3p5T11+p6T12 |
| 31 | (1+10T+119T2+636T3+119pT4+10p2T5+p3T6)2 |
| 37 | (1+4T+pT2)6 |
| 41 | 1+8T−11T2−456T3−1674T4+3848T5+78417T6+3848pT7−1674p2T8−456p3T9−11p4T10+8p5T11+p6T12 |
| 43 | 1+7T−41T2−620T3−145T4+15013T5+110738T6+15013pT7−145p2T8−620p3T9−41p4T10+7p5T11+p6T12 |
| 47 | 1−7T−pT2+30T3+2247T4+14009T5−210546T6+14009pT7+2247p2T8+30p3T9−p5T10−7p5T11+p6T12 |
| 53 | 1+26T+309T2+3038T3+30686T4+255066T5+1835857T6+255066pT7+30686p2T8+3038p3T9+309p4T10+26p5T11+p6T12 |
| 59 | 1+2T−157T2−2pT3+15982T4+4386T5−1084105T6+4386pT7+15982p2T8−2p4T9−157p4T10+2p5T11+p6T12 |
| 61 | 1+3T−2T2−851T3−1876T4+11999T5+608456T6+11999pT7−1876p2T8−851p3T9−2p4T10+3p5T11+p6T12 |
| 67 | 1+6T−111T2−294T3+8232T4−4818T5−662137T6−4818pT7+8232p2T8−294p3T9−111p4T10+6p5T11+p6T12 |
| 71 | 1−7T−163T2+458T3+23095T4−30411T5−1779370T6−30411pT7+23095p2T8+458p3T9−163p4T10−7p5T11+p6T12 |
| 73 | 1+6T−59T2+562T3+2066T4−50866T5−72619T6−50866pT7+2066p2T8+562p3T9−59p4T10+6p5T11+p6T12 |
| 79 | 1−6T−101T2+386T3+4322T4+9682T5−356533T6+9682pT7+4322p2T8+386p3T9−101p4T10−6p5T11+p6T12 |
| 83 | (1+19T+353T2+3326T3+353pT4+19p2T5+p3T6)2 |
| 89 | 1−37T+678T2−9571T3+120056T4−1268769T5+12038284T6−1268769pT7+120056p2T8−9571p3T9+678p4T10−37p5T11+p6T12 |
| 97 | 1+T−129T2+800T3+4625T4−59361T5+288862T6−59361pT7+4625p2T8+800p3T9−129p4T10+p5T11+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
−5.13133271252072893674056400176, −5.01349495829686994755413251962, −4.75070612325496286856383644143, −4.73377002026592167999445455014, −4.56702763736243982602026955342, −4.44499346451817828377566800751, −4.02640903312572756686300734110, −3.90413429356284829533504128080, −3.87563390138396654404110419693, −3.52641340079845831945129214393, −3.27431575724861398457916195238, −3.27069710822891362442496794362, −3.16018714334800215031856824368, −2.98950970044281064632847971613, −2.52816104740058299561546715425, −2.38198106122704904174628813912, −1.86458780566143654213150004210, −1.82586245945233310526662749245, −1.76098107566736558313097400787, −1.69273634176523288362704593174, −1.57646064633531466723583618273, −1.43421892080224852458218805044, −1.09728472047749531136290061185, −0.893708026341500811278208609820, −0.28421253234324838221673942225,
0.28421253234324838221673942225, 0.893708026341500811278208609820, 1.09728472047749531136290061185, 1.43421892080224852458218805044, 1.57646064633531466723583618273, 1.69273634176523288362704593174, 1.76098107566736558313097400787, 1.82586245945233310526662749245, 1.86458780566143654213150004210, 2.38198106122704904174628813912, 2.52816104740058299561546715425, 2.98950970044281064632847971613, 3.16018714334800215031856824368, 3.27069710822891362442496794362, 3.27431575724861398457916195238, 3.52641340079845831945129214393, 3.87563390138396654404110419693, 3.90413429356284829533504128080, 4.02640903312572756686300734110, 4.44499346451817828377566800751, 4.56702763736243982602026955342, 4.73377002026592167999445455014, 4.75070612325496286856383644143, 5.01349495829686994755413251962, 5.13133271252072893674056400176
Plot not available for L-functions of degree greater than 10.