L(s) = 1 | − 2·3-s − 6·7-s − 3·9-s − 2·11-s − 14·13-s − 10·17-s + 6·19-s + 12·21-s − 2·23-s + 10·27-s − 2·29-s + 8·31-s + 4·33-s − 4·37-s + 28·39-s + 4·41-s − 4·43-s − 4·47-s − 2·49-s + 20·51-s + 14·53-s − 12·57-s − 2·59-s + 10·61-s + 18·63-s − 42·67-s + 4·69-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 2.26·7-s − 9-s − 0.603·11-s − 3.88·13-s − 2.42·17-s + 1.37·19-s + 2.61·21-s − 0.417·23-s + 1.92·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s − 0.657·37-s + 4.48·39-s + 0.624·41-s − 0.609·43-s − 0.583·47-s − 2/7·49-s + 2.80·51-s + 1.92·53-s − 1.58·57-s − 0.260·59-s + 1.28·61-s + 2.26·63-s − 5.13·67-s + 0.481·69-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) |
= |
0 |
L(21) |
= |
0 |
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+2T+7T2+10T3+25T4+32T5+86T6+32pT7+25p2T8+10p3T9+7p4T10+2p5T11+p6T12 |
| 7 | 1+6T+38T2+148T3+82pT4+1674T5+4994T6+1674pT7+82p3T8+148p3T9+38p4T10+6p5T11+p6T12 |
| 11 | 1+2T+13T2+34T3+245T4+368T5+2322T6+368pT7+245p2T8+34p3T9+13p4T10+2p5T11+p6T12 |
| 13 | 1+14T+105T2+558T3+2669T4+924pT5+47674T6+924p2T7+2669p2T8+558p3T9+105p4T10+14p5T11+p6T12 |
| 17 | 1+10T+4pT2+276T3+1308T4+6202T5+30858T6+6202pT7+1308p2T8+276p3T9+4p5T10+10p5T11+p6T12 |
| 23 | 1+2T+77T2+186T3+2931T4+6760T5+77966T6+6760pT7+2931p2T8+186p3T9+77p4T10+2p5T11+p6T12 |
| 29 | 1+2T+79T2−102T3+3027T4−8324T5+101530T6−8324pT7+3027p2T8−102p3T9+79p4T10+2p5T11+p6T12 |
| 31 | 1−8T+148T2−920T3+10167T4−50192T5+403176T6−50192pT7+10167p2T8−920p3T9+148p4T10−8p5T11+p6T12 |
| 37 | 1+4T+100T2+436T3+6119T4+22616T5+267960T6+22616pT7+6119p2T8+436p3T9+100p4T10+4p5T11+p6T12 |
| 41 | 1−4T+188T2−676T3+16775T4−50488T5+876504T6−50488pT7+16775p2T8−676p3T9+188p4T10−4p5T11+p6T12 |
| 43 | 1+4T+141T2+712T3+11185T4+49420T5+594874T6+49420pT7+11185p2T8+712p3T9+141p4T10+4p5T11+p6T12 |
| 47 | 1+4T+243T2+816T3+26239T4+71836T5+1599386T6+71836pT7+26239p2T8+816p3T9+243p4T10+4p5T11+p6T12 |
| 53 | 1−14T+249T2−2462T3+28173T4−220540T5+1887962T6−220540pT7+28173p2T8−2462p3T9+249p4T10−14p5T11+p6T12 |
| 59 | 1+2T+89T2+114T3+9599T4+8120T5+549326T6+8120pT7+9599p2T8+114p3T9+89p4T10+2p5T11+p6T12 |
| 61 | 1−10T+233T2−1710T3+27173T4−171840T5+2075338T6−171840pT7+27173p2T8−1710p3T9+233p4T10−10p5T11+p6T12 |
| 67 | 1+42T+1067T2+18970T3+262101T4+2890456T5+26146286T6+2890456pT7+262101p2T8+18970p3T9+1067p4T10+42p5T11+p6T12 |
| 71 | 1+8T+364T2+2736T3+58431T4+381256T5+5343112T6+381256pT7+58431p2T8+2736p3T9+364p4T10+8p5T11+p6T12 |
| 73 | 1−2T+276T2−1356T3+33692T4−238482T5+2758666T6−238482pT7+33692p2T8−1356p3T9+276p4T10−2p5T11+p6T12 |
| 79 | 1+20T+590T2+7996T3+129599T4+1270184T5+14099364T6+1270184pT7+129599p2T8+7996p3T9+590p4T10+20p5T11+p6T12 |
| 83 | 1+16T+510T2+6064T3+105831T4+961376T5+11690948T6+961376pT7+105831p2T8+6064p3T9+510p4T10+16p5T11+p6T12 |
| 89 | 1−32T+656T2−9816T3+119487T4−1272488T5+12355776T6−1272488pT7+119487p2T8−9816p3T9+656p4T10−32p5T11+p6T12 |
| 97 | 1+40T+1060T2+20000T3+303631T4+3806872T5+40585688T6+3806872pT7+303631p2T8+20000p3T9+1060p4T10+40p5T11+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
−4.86137399173361902895522723888, −4.70946292907760828137527104972, −4.48776180496958968650406602687, −4.38815158110018228542326278594, −4.28597972795277125188036131091, −4.20519312046796208485193517784, −4.16564523352090037701538168768, −3.62335770986501353970037250780, −3.60768974297536016359389490388, −3.46172131490042659473916346950, −3.32255847924512262847848311483, −3.14691357573419607020384144070, −3.14154801717236225919931703717, −2.65373226129974490529725176771, −2.60743172684876293927640633048, −2.58811194199752644047495856550, −2.49292933297905422831706578576, −2.49201948616089429876807818676, −2.42444030761923594410837368434, −1.99596305248834091842092385569, −1.66661355496970272230647656251, −1.41300228621107213104338394486, −1.40249609117630693939587679186, −1.02054784796983291449167355497, −0.966082243953473213882923843641, 0, 0, 0, 0, 0, 0,
0.966082243953473213882923843641, 1.02054784796983291449167355497, 1.40249609117630693939587679186, 1.41300228621107213104338394486, 1.66661355496970272230647656251, 1.99596305248834091842092385569, 2.42444030761923594410837368434, 2.49201948616089429876807818676, 2.49292933297905422831706578576, 2.58811194199752644047495856550, 2.60743172684876293927640633048, 2.65373226129974490529725176771, 3.14154801717236225919931703717, 3.14691357573419607020384144070, 3.32255847924512262847848311483, 3.46172131490042659473916346950, 3.60768974297536016359389490388, 3.62335770986501353970037250780, 4.16564523352090037701538168768, 4.20519312046796208485193517784, 4.28597972795277125188036131091, 4.38815158110018228542326278594, 4.48776180496958968650406602687, 4.70946292907760828137527104972, 4.86137399173361902895522723888
Plot not available for L-functions of degree greater than 10.