L(s) = 1 | + 2·5-s − 42·7-s − 51·9-s + 66·11-s − 6·13-s − 14·17-s − 80·19-s − 254·23-s − 263·25-s − 30·27-s + 132·29-s + 52·31-s − 84·35-s − 518·37-s + 486·41-s − 428·43-s − 102·45-s − 790·47-s + 1.02e3·49-s − 40·53-s + 132·55-s − 436·59-s + 1.03e3·61-s + 2.14e3·63-s − 12·65-s − 562·67-s − 2.47e3·71-s + ⋯ |
L(s) = 1 | + 0.178·5-s − 2.26·7-s − 1.88·9-s + 1.80·11-s − 0.128·13-s − 0.199·17-s − 0.965·19-s − 2.30·23-s − 2.10·25-s − 0.213·27-s + 0.845·29-s + 0.301·31-s − 0.405·35-s − 2.30·37-s + 1.85·41-s − 1.51·43-s − 0.337·45-s − 2.45·47-s + 3·49-s − 0.103·53-s + 0.323·55-s − 0.962·59-s + 2.17·61-s + 4.28·63-s − 0.0228·65-s − 1.02·67-s − 4.13·71-s + ⋯ |
Λ(s)=(=((224⋅76⋅116)s/2ΓC(s)6L(s)Λ(4−s)
Λ(s)=(=((224⋅76⋅116)s/2ΓC(s+3/2)6L(s)Λ(1−s)
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | (1+pT)6 |
| 11 | (1−pT)6 |
good | 3 | 1+17pT2+10pT3+521pT4+818pT5+33458T6+818p4T7+521p7T8+10p10T9+17p13T10+p18T12 |
| 5 | 1−2T+267T2+78T3+28059T4−30628T5+2913874T6−30628p3T7+28059p6T8+78p9T9+267p12T10−2p15T11+p18T12 |
| 13 | 1+6T+7226T2−12986T3+21595335T4−212545140T5+46341531436T6−212545140p3T7+21595335p6T8−12986p9T9+7226p12T10+6p15T11+p18T12 |
| 17 | 1+14T+13486T2+38686pT3+90635967T4+7350497812T5+447812918500T6+7350497812p3T7+90635967p6T8+38686p10T9+13486p12T10+14p15T11+p18T12 |
| 19 | 1+80T+20430T2+808800T3+107899847T4−1308862768T5+219419401156T6−1308862768p3T7+107899847p6T8+808800p9T9+20430p12T10+80p15T11+p18T12 |
| 23 | 1+254T+70315T2+12243950T3+2051483067T4+263880838960T5+32746801812082T6+263880838960p3T7+2051483067p6T8+12243950p9T9+70315p12T10+254p15T11+p18T12 |
| 29 | 1−132T+99898T2−9991364T3+4823385367T4−405808479304T5+147113503501772T6−405808479304p3T7+4823385367p6T8−9991364p9T9+99898p12T10−132p15T11+p18T12 |
| 31 | 1−52T+89883T2+2087062T3+3252254323T4+377234281626T5+87166967790482T6+377234281626p3T7+3252254323p6T8+2087062p9T9+89883p12T10−52p15T11+p18T12 |
| 37 | 1+14pT+269207T2+85165306T3+28898877291T4+7179634256120T5+1864618903150618T6+7179634256120p3T7+28898877291p6T8+85165306p9T9+269207p12T10+14p16T11+p18T12 |
| 41 | 1−486T+323230T2−119780726T3+46993119439T4−14376699282932T5+4122837682268676T6−14376699282932p3T7+46993119439p6T8−119780726p9T9+323230p12T10−486p15T11+p18T12 |
| 43 | 1+428T+393842T2+133211668T3+71876415271T4+19044270193336T5+7379742977723868T6+19044270193336p3T7+71876415271p6T8+133211668p9T9+393842p12T10+428p15T11+p18T12 |
| 47 | 1+790T+642258T2+333678938T3+3492337457pT4+62727363087012T5+22537508139391772T6+62727363087012p3T7+3492337457p7T8+333678938p9T9+642258p12T10+790p15T11+p18T12 |
| 53 | 1+40T+208850T2−40763096T3+24540166119T4−1549444871536T5+5015035290868764T6−1549444871536p3T7+24540166119p6T8−40763096p9T9+208850p12T10+40p15T11+p18T12 |
| 59 | 1+436T+945043T2+344755578T3+414850671851T4+125541076038526T5+108238520466107794T6+125541076038526p3T7+414850671851p6T8+344755578p9T9+945043p12T10+436p15T11+p18T12 |
| 61 | 1−1034T+1213638T2−827989170T3+607908506471T4−325318914419660T5+178537931833608916T6−325318914419660p3T7+607908506471p6T8−827989170p9T9+1213638p12T10−1034p15T11+p18T12 |
| 67 | 1+562T+1314091T2+416909682T3+681384395571T4+123413911558312T5+227606452205865378T6+123413911558312p3T7+681384395571p6T8+416909682p9T9+1314091p12T10+562p15T11+p18T12 |
| 71 | 1+2474T+4156019T2+4921377706T3+4708740035227T4+3654320624597408T5+474943847471938p2T6+3654320624597408p3T7+4708740035227p6T8+4921377706p9T9+4156019p12T10+2474p15T11+p18T12 |
| 73 | 1+902T+1464166T2+1154693750T3+1041995682511T4+711491269010964T5+487603872205165556T6+711491269010964p3T7+1041995682511p6T8+1154693750p9T9+1464166p12T10+902p15T11+p18T12 |
| 79 | 1+1636T+2498302T2+33331924pT3+2469987735583T4+1899577945348872T5+1461465898798234628T6+1899577945348872p3T7+2469987735583p6T8+33331924p10T9+2498302p12T10+1636p15T11+p18T12 |
| 83 | 1+3016T+6643466T2+10104768056T3+12515177745655T4+12402952946307472T5+10338749994204949036T6+12402952946307472p3T7+12515177745655p6T8+10104768056p9T9+6643466p12T10+3016p15T11+p18T12 |
| 89 | 1−1750T+4497595T2−5583770934T3+8155725959611T4−7522990771478308T5+7722629637376363346T6−7522990771478308p3T7+8155725959611p6T8−5583770934p9T9+4497595p12T10−1750p15T11+p18T12 |
| 97 | 1+1250T+2518379T2+1593791166T3+1571798248867T4−304189816010912T5+214623295214286178T6−304189816010912p3T7+1571798248867p6T8+1593791166p9T9+2518379p12T10+1250p15T11+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
−5.33504046785270954536945638056, −5.08314460241253563282372964997, −4.71145446404979194842971755679, −4.57492196384778921571260815736, −4.56621241759995096655779570342, −4.30678762169290093554357319575, −4.24571104849646553433490848061, −4.06521302041563926912556724862, −3.71027064984486985497784032491, −3.70009535075218910414393666736, −3.56554647511755633936860602897, −3.42600235738734403691786764798, −3.42538556704132538781174499830, −2.86655449367963239661397147184, −2.86478931064933179518261142187, −2.72397453781365153282787214351, −2.50227899117176488301048214467, −2.39947265662062544018605350874, −2.34064403223413215077461014651, −1.71170853773646993767696594174, −1.65508884688342643867650367389, −1.51460663808947003980653283875, −1.45131547181095145990516625340, −1.08742501638657976948156490682, −0.918259867719945211447512254193, 0, 0, 0, 0, 0, 0,
0.918259867719945211447512254193, 1.08742501638657976948156490682, 1.45131547181095145990516625340, 1.51460663808947003980653283875, 1.65508884688342643867650367389, 1.71170853773646993767696594174, 2.34064403223413215077461014651, 2.39947265662062544018605350874, 2.50227899117176488301048214467, 2.72397453781365153282787214351, 2.86478931064933179518261142187, 2.86655449367963239661397147184, 3.42538556704132538781174499830, 3.42600235738734403691786764798, 3.56554647511755633936860602897, 3.70009535075218910414393666736, 3.71027064984486985497784032491, 4.06521302041563926912556724862, 4.24571104849646553433490848061, 4.30678762169290093554357319575, 4.56621241759995096655779570342, 4.57492196384778921571260815736, 4.71145446404979194842971755679, 5.08314460241253563282372964997, 5.33504046785270954536945638056
Plot not available for L-functions of degree greater than 10.