L(s) = 1 | + 5·9-s − 6·19-s − 34·29-s + 4·31-s + 12·41-s + 17·49-s − 30·59-s + 16·61-s − 8·71-s − 4·79-s + 2·81-s − 12·89-s − 24·101-s − 38·109-s − 66·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 57·169-s − 30·171-s + 173-s + ⋯ |
L(s) = 1 | + 5/3·9-s − 1.37·19-s − 6.31·29-s + 0.718·31-s + 1.87·41-s + 17/7·49-s − 3.90·59-s + 2.04·61-s − 0.949·71-s − 0.450·79-s + 2/9·81-s − 1.27·89-s − 2.38·101-s − 3.63·109-s − 6·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.38·169-s − 2.29·171-s + 0.0760·173-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) |
≈ |
3.661780061 |
L(21) |
≈ |
3.661780061 |
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 | 1−5T2+23T4−86T6+23p2T8−5p4T10+p6T12 |
| 7 | 1−17T2+211T4−1718T6+211p2T8−17p4T10+p6T12 |
| 11 | (1+pT2)6 |
| 13 | 1−57T2+1487T4−23774T6+1487p2T8−57p4T10+p6T12 |
| 17 | 1−69T2+2147T4−42926T6+2147p2T8−69p4T10+p6T12 |
| 23 | 1−73T2+3107T4−85926T6+3107p2T8−73p4T10+p6T12 |
| 29 | (1+17T+171T2+1110T3+171pT4+17p2T5+p3T6)2 |
| 31 | (1−2T+45T2+4T3+45pT4−2p2T5+p3T6)2 |
| 37 | 1−158T2+11191T4−496772T6+11191p2T8−158p4T10+p6T12 |
| 41 | (1−6T+71T2−436T3+71pT4−6p2T5+p3T6)2 |
| 43 | 1−174T2+13991T4−717764T6+13991p2T8−174p4T10+p6T12 |
| 47 | 1−106T2+5423T4−232716T6+5423p2T8−106p4T10+p6T12 |
| 53 | 1−209T2+20271T4−1269614T6+20271p2T8−209p4T10+p6T12 |
| 59 | (1+15T+149T2+986T3+149pT4+15p2T5+p3T6)2 |
| 61 | (1−8T+179T2−912T3+179pT4−8p2T5+p3T6)2 |
| 67 | 1−37T2+7511T4−169110T6+7511p2T8−37p4T10+p6T12 |
| 71 | (1+4T+21T2−456T3+21pT4+4p2T5+p3T6)2 |
| 73 | 1−229T2+27155T4−30366pT6+27155p2T8−229p4T10+p6T12 |
| 79 | (1+2T+213T2+284T3+213pT4+2p2T5+p3T6)2 |
| 83 | 1−366T2+60407T4−6127364T6+60407p2T8−366p4T10+p6T12 |
| 89 | (1+6T+215T2+884T3+215pT4+6p2T5+p3T6)2 |
| 97 | 1−358T2+62671T4−7159060T6+62671p2T8−358p4T10+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
−4.28796108798481282090230690788, −4.23071940256840709222236083897, −4.18499771242696888959210147481, −4.05960019480092712013542013190, −3.79803893725793442786446647294, −3.69165310339962767369560678964, −3.57495112561867474205829030122, −3.56428376748819566317412804000, −3.16697465462797393106678710151, −3.15188261646090351558425730707, −2.77814777577673496579588190349, −2.71488540180933286473424904123, −2.45743926949288468244277081776, −2.33068654746527686165677783329, −2.30883090086229616369517314445, −2.30617675650838253723635086167, −1.68738148471675665015981328022, −1.64587361621402961688956936471, −1.56372697768125446551306778271, −1.41903135605133277498661569868, −1.27309282765583189636570105611, −1.19105542465414532992862668459, −0.39143926442762275536399869671, −0.37173644232868418529930824677, −0.34400869778399379051097310846,
0.34400869778399379051097310846, 0.37173644232868418529930824677, 0.39143926442762275536399869671, 1.19105542465414532992862668459, 1.27309282765583189636570105611, 1.41903135605133277498661569868, 1.56372697768125446551306778271, 1.64587361621402961688956936471, 1.68738148471675665015981328022, 2.30617675650838253723635086167, 2.30883090086229616369517314445, 2.33068654746527686165677783329, 2.45743926949288468244277081776, 2.71488540180933286473424904123, 2.77814777577673496579588190349, 3.15188261646090351558425730707, 3.16697465462797393106678710151, 3.56428376748819566317412804000, 3.57495112561867474205829030122, 3.69165310339962767369560678964, 3.79803893725793442786446647294, 4.05960019480092712013542013190, 4.18499771242696888959210147481, 4.23071940256840709222236083897, 4.28796108798481282090230690788
Plot not available for L-functions of degree greater than 10.