L(s) = 1 | − 2-s + 2·4-s + 30·5-s + 47·7-s − 16·8-s − 30·10-s − 81·11-s + 169·13-s − 47·14-s − 53·16-s + 17-s + 116·19-s + 60·20-s + 81·22-s + 52·23-s + 525·25-s − 169·26-s + 94·28-s − 174·29-s + 340·31-s + 21·32-s − 34-s + 1.41e3·35-s + 332·37-s − 116·38-s − 480·40-s + 616·41-s + ⋯ |
L(s) = 1 | − 0.353·2-s + 1/4·4-s + 2.68·5-s + 2.53·7-s − 0.707·8-s − 0.948·10-s − 2.22·11-s + 3.60·13-s − 0.897·14-s − 0.828·16-s + 0.0142·17-s + 1.40·19-s + 0.670·20-s + 0.784·22-s + 0.471·23-s + 21/5·25-s − 1.27·26-s + 0.634·28-s − 1.11·29-s + 1.96·31-s + 0.116·32-s − 0.00504·34-s + 6.80·35-s + 1.47·37-s − 0.495·38-s − 1.89·40-s + 2.34·41-s + ⋯ |
Λ(s)=(=((312⋅56⋅296)s/2ΓC(s)6L(s)Λ(4−s)
Λ(s)=(=((312⋅56⋅296)s/2ΓC(s+3/2)6L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
138.5304385 |
L(21) |
≈ |
138.5304385 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | (1−pT)6 |
| 29 | (1+pT)6 |
good | 2 | 1+T−T2+13T3+21p2T4+37pT5−23p2T6+37p4T7+21p8T8+13p9T9−p12T10+p15T11+p18T12 |
| 7 | 1−47T+242pT2−48633T3+1269399T4−27084898T5+536364308T6−27084898p3T7+1269399p6T8−48633p9T9+242p13T10−47p15T11+p18T12 |
| 11 | 1+81T+7285T2+400470T3+21225287T4+884246505T5+35457827646T6+884246505p3T7+21225287p6T8+400470p9T9+7285p12T10+81p15T11+p18T12 |
| 13 | 1−p2T+22166T2−2003717T3+150322519T4−9100077590T5+467902766548T6−9100077590p3T7+150322519p6T8−2003717p9T9+22166p12T10−p17T11+p18T12 |
| 17 | 1−T+22636T2+54303T3+231271391T4+914499890T5+1416714172968T6+914499890p3T7+231271391p6T8+54303p9T9+22636p12T10−p15T11+p18T12 |
| 19 | 1−116T+31902T2−2590388T3+453795591T4−30145107120T5+209154567628pT6−30145107120p3T7+453795591p6T8−2590388p9T9+31902p12T10−116p15T11+p18T12 |
| 23 | 1−52T+24683T2−1496510T3+282119759T4−27844771778T5+2488866804234T6−27844771778p3T7+282119759p6T8−1496510p9T9+24683p12T10−52p15T11+p18T12 |
| 31 | 1−340T+181850T2−45373684T3+13429117455T4−2551173914240T5+530229057162156T6−2551173914240p3T7+13429117455p6T8−45373684p9T9+181850p12T10−340p15T11+p18T12 |
| 37 | 1−332T+292071T2−76574622T3+36281439399T4−7392542582574T5+2436574283519458T6−7392542582574p3T7+36281439399p6T8−76574622p9T9+292071p12T10−332p15T11+p18T12 |
| 41 | 1−616T+435339T2−176922950T3+74464996755T4−22341033150810T5+6778583702035218T6−22341033150810p3T7+74464996755p6T8−176922950p9T9+435339p12T10−616p15T11+p18T12 |
| 43 | 1−334T+430715T2−123508878T3+79745074707T4−18904283647592T5+8240498797811330T6−18904283647592p3T7+79745074707p6T8−123508878p9T9+430715p12T10−334p15T11+p18T12 |
| 47 | 1−85T+6534pT2−27129259T3+46333879375T4−4270390938886T5+5227445483301100T6−4270390938886p3T7+46333879375p6T8−27129259p9T9+6534p13T10−85p15T11+p18T12 |
| 53 | 1−850T+1116583T2−652304118T3+466050574283T4−197142654656416T5+95871162711193914T6−197142654656416p3T7+466050574283p6T8−652304118p9T9+1116583p12T10−850p15T11+p18T12 |
| 59 | 1−758T+1182306T2−633358186T3+565678561863T4−231953762059260T5+150395021848589820T6−231953762059260p3T7+565678561863p6T8−633358186p9T9+1182306p12T10−758p15T11+p18T12 |
| 61 | 1+36T+1049294T2+61806916T3+508396683767T4+31304512521192T5+145855929933536228T6+31304512521192p3T7+508396683767p6T8+61806916p9T9+1049294p12T10+36p15T11+p18T12 |
| 67 | 1−939T+1083468T2−677838757T3+553019898167T4−325001005895942T5+213102331026860744T6−325001005895942p3T7+553019898167p6T8−677838757p9T9+1083468p12T10−939p15T11+p18T12 |
| 71 | 1−1388T+1656622T2−1864287268T3+1492186295791T4−1084857473044696T5+736165848562113508T6−1084857473044696p3T7+1492186295791p6T8−1864287268p9T9+1656622p12T10−1388p15T11+p18T12 |
| 73 | 1−1708T+2388831T2−2092640994T3+1770401083215T4−1181456884941858T5+815155934532229474T6−1181456884941858p3T7+1770401083215p6T8−2092640994p9T9+2388831p12T10−1708p15T11+p18T12 |
| 79 | 1−1250T+2597586T2−2405321470T3+2913257383423T4−2078370256058300T5+1851933970653265820T6−2078370256058300p3T7+2913257383423p6T8−2405321470p9T9+2597586p12T10−1250p15T11+p18T12 |
| 83 | 1−748T+1762807T2−1534149802T3+1864138806963T4−1377438064407974T5+1341134023818679450T6−1377438064407974p3T7+1864138806963p6T8−1534149802p9T9+1762807p12T10−748p15T11+p18T12 |
| 89 | 1+1099T+3491726T2+39764275pT3+5580205971935T4+4728799528650002T5+5098038721965740676T6+4728799528650002p3T7+5580205971935p6T8+39764275p10T9+3491726p12T10+1099p15T11+p18T12 |
| 97 | 1+22T+2535151T2+160798614T3+3549398875803T4+639134561324052T5+3783183972148295146T6+639134561324052p3T7+3549398875803p6T8+160798614p9T9+2535151p12T10+22p15T11+p18T12 |
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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
−4.98370350037198139186335636093, −4.48836606616365249344251319079, −4.23982014287875410189527529179, −4.14133788314981326275958973523, −4.12863180188214425155256905578, −4.05559119200299172787473409784, −3.70167586759197876619678908477, −3.39574750201322704650397877823, −3.31521218844110823993166677023, −3.11957154074524617611821767225, −2.94054107922054638998877885450, −2.64022643579589217851844746362, −2.43289973869564801422534187091, −2.43089866597579298983439848808, −2.23978371928515366652093229612, −2.23571143552221040748030346376, −1.72574226942821449778547250817, −1.72167621806608542770626682447, −1.62451858496124770010980823076, −1.34101284007026192823134828260, −0.865463109453529623537002492798, −0.819810116994125049973339714016, −0.76591844627920522071778016409, −0.62430851322047967491468507308, −0.58070235365664815954997028375,
0.58070235365664815954997028375, 0.62430851322047967491468507308, 0.76591844627920522071778016409, 0.819810116994125049973339714016, 0.865463109453529623537002492798, 1.34101284007026192823134828260, 1.62451858496124770010980823076, 1.72167621806608542770626682447, 1.72574226942821449778547250817, 2.23571143552221040748030346376, 2.23978371928515366652093229612, 2.43089866597579298983439848808, 2.43289973869564801422534187091, 2.64022643579589217851844746362, 2.94054107922054638998877885450, 3.11957154074524617611821767225, 3.31521218844110823993166677023, 3.39574750201322704650397877823, 3.70167586759197876619678908477, 4.05559119200299172787473409784, 4.12863180188214425155256905578, 4.14133788314981326275958973523, 4.23982014287875410189527529179, 4.48836606616365249344251319079, 4.98370350037198139186335636093
Plot not available for L-functions of degree greater than 10.