L(s) = 1 | − 2·3-s − 2·7-s − 2·9-s − 3·11-s − 3·13-s − 2·17-s − 6·19-s + 4·21-s − 4·23-s + 2·27-s + 7·29-s − 5·31-s + 6·33-s + 6·39-s + 11·41-s + 7·43-s − 20·47-s − 20·49-s + 4·51-s − 7·53-s + 12·57-s + 4·59-s + 13·61-s + 4·63-s − 25·67-s + 8·69-s − 29·71-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.755·7-s − 2/3·9-s − 0.904·11-s − 0.832·13-s − 0.485·17-s − 1.37·19-s + 0.872·21-s − 0.834·23-s + 0.384·27-s + 1.29·29-s − 0.898·31-s + 1.04·33-s + 0.960·39-s + 1.71·41-s + 1.06·43-s − 2.91·47-s − 2.85·49-s + 0.560·51-s − 0.961·53-s + 1.58·57-s + 0.520·59-s + 1.66·61-s + 0.503·63-s − 3.05·67-s + 0.963·69-s − 3.44·71-s + ⋯ |
Λ(s)=(=((224⋅512⋅196)s/2ΓC(s)6L(s)Λ(2−s)
Λ(s)=(=((224⋅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+2pT2+14T3+8pT4+40T5+83T6+40pT7+8p3T8+14p3T9+2p5T10+2p5T11+p6T12 |
| 7 | 1+2T+24T2+54T3+318T4+594T5+2791T6+594pT7+318p2T8+54p3T9+24p4T10+2p5T11+p6T12 |
| 11 | 1+3T+14T2+50T3+433T4+915T5+3400T6+915pT7+433p2T8+50p3T9+14p4T10+3p5T11+p6T12 |
| 13 | 1+3T+35T2+134T3+790T4+2388T5+13141T6+2388pT7+790p2T8+134p3T9+35p4T10+3p5T11+p6T12 |
| 17 | 1+2T+40T2−12T3+422T4−2878T5+309T6−2878pT7+422p2T8−12p3T9+40p4T10+2p5T11+p6T12 |
| 23 | 1+4T+96T2+346T3+4540T4+14008T5+131019T6+14008pT7+4540p2T8+346p3T9+96p4T10+4p5T11+p6T12 |
| 29 | 1−7T+101T2−680T3+6104T4−32894T5+218211T6−32894pT7+6104p2T8−680p3T9+101p4T10−7p5T11+p6T12 |
| 31 | 1+5T+116T2+166T3+4207T4−9815T5+97168T6−9815pT7+4207p2T8+166p3T9+116p4T10+5p5T11+p6T12 |
| 37 | 1+14T2+189T3+1675T4+1758T5+75767T6+1758pT7+1675p2T8+189p3T9+14p4T10+p6T12 |
| 41 | 1−11T+162T2−1616T3+14575T4−107273T5+784764T6−107273pT7+14575p2T8−1616p3T9+162p4T10−11p5T11+p6T12 |
| 43 | 1−7T+pT2−530T3+3441T4−21767T5+216350T6−21767pT7+3441p2T8−530p3T9+p5T10−7p5T11+p6T12 |
| 47 | 1+20T+294T2+2363T3+14371T4+39728T5+169611T6+39728pT7+14371p2T8+2363p3T9+294p4T10+20p5T11+p6T12 |
| 53 | 1+7T+286T2+1674T3+35709T4+168565T5+2474973T6+168565pT7+35709p2T8+1674p3T9+286p4T10+7p5T11+p6T12 |
| 59 | 1−4T+261T2−825T3+32193T4−80901T5+2380506T6−80901pT7+32193p2T8−825p3T9+261p4T10−4p5T11+p6T12 |
| 61 | 1−13T+246T2−2418T3+29973T4−235869T5+2276056T6−235869pT7+29973p2T8−2418p3T9+246p4T10−13p5T11+p6T12 |
| 67 | 1+25T+505T2+6772T3+80896T4+773578T5+6921907T6+773578pT7+80896p2T8+6772p3T9+505p4T10+25p5T11+p6T12 |
| 71 | 1+29T+603T2+8526T3+102909T4+1014029T5+9209494T6+1014029pT7+102909p2T8+8526p3T9+603p4T10+29p5T11+p6T12 |
| 73 | 1+19T+505T2+6558T3+98292T4+932744T5+9748099T6+932744pT7+98292p2T8+6558p3T9+505p4T10+19p5T11+p6T12 |
| 79 | 1+28T+659T2+10371T3+145815T4+1597403T5+15814250T6+1597403pT7+145815p2T8+10371p3T9+659p4T10+28p5T11+p6T12 |
| 83 | 1−15T+496T2−5598T3+100571T4−875935T5+11004272T6−875935pT7+100571p2T8−5598p3T9+496p4T10−15p5T11+p6T12 |
| 89 | 1+12T+317T2+3161T3+56077T4+457091T5+6094922T6+457091pT7+56077p2T8+3161p3T9+317p4T10+12p5T11+p6T12 |
| 97 | 1+13T+448T2+3788T3+80619T4+494279T5+9036752T6+494279pT7+80619p2T8+3788p3T9+448p4T10+13p5T11+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
−4.36676476902041552046454200249, −4.36230357836642365054978148586, −4.09178084112276884074677868663, −4.03074601243493776864288979693, −3.93676183210762841659172416940, −3.70110372760700890770195139898, −3.68635604612165284776725692306, −3.48603336762707828547689857725, −3.31351080060677559934017819840, −3.09205691665947725491564995582, −2.97845914929348359928696872621, −2.87159552950521352379257621528, −2.85141467397530161618638254940, −2.49471826207213807371905269683, −2.45372036117674958744443515264, −2.42255912755887699568598329853, −2.35304463057718885774617324629, −2.23343770572331742068642760893, −1.67391833147898922553787888617, −1.55965799822884664057354508439, −1.51139022598239703566608084162, −1.40736898170938260515380454094, −1.28045491613866060849014307113, −1.09765912467188099419211378574, −0.73001629642707044811614098368, 0, 0, 0, 0, 0, 0,
0.73001629642707044811614098368, 1.09765912467188099419211378574, 1.28045491613866060849014307113, 1.40736898170938260515380454094, 1.51139022598239703566608084162, 1.55965799822884664057354508439, 1.67391833147898922553787888617, 2.23343770572331742068642760893, 2.35304463057718885774617324629, 2.42255912755887699568598329853, 2.45372036117674958744443515264, 2.49471826207213807371905269683, 2.85141467397530161618638254940, 2.87159552950521352379257621528, 2.97845914929348359928696872621, 3.09205691665947725491564995582, 3.31351080060677559934017819840, 3.48603336762707828547689857725, 3.68635604612165284776725692306, 3.70110372760700890770195139898, 3.93676183210762841659172416940, 4.03074601243493776864288979693, 4.09178084112276884074677868663, 4.36230357836642365054978148586, 4.36676476902041552046454200249
Plot not available for L-functions of degree greater than 10.