L(s) = 1 | − 9·3-s − 6·5-s − 6·7-s + 36·9-s − 6·11-s − 15·13-s + 54·15-s − 9·17-s − 18·19-s + 54·21-s − 18·23-s + 18·25-s − 80·27-s − 3·29-s + 3·31-s + 54·33-s + 36·35-s − 24·37-s + 135·39-s − 15·41-s + 21·43-s − 216·45-s − 21·47-s + 30·49-s + 81·51-s − 27·53-s + 36·55-s + ⋯ |
L(s) = 1 | − 5.19·3-s − 2.68·5-s − 2.26·7-s + 12·9-s − 1.80·11-s − 4.16·13-s + 13.9·15-s − 2.18·17-s − 4.12·19-s + 11.7·21-s − 3.75·23-s + 18/5·25-s − 15.3·27-s − 0.557·29-s + 0.538·31-s + 9.40·33-s + 6.08·35-s − 3.94·37-s + 21.6·39-s − 2.34·41-s + 3.20·43-s − 32.1·45-s − 3.06·47-s + 30/7·49-s + 11.3·51-s − 3.70·53-s + 4.85·55-s + ⋯ |
Λ(s)=(=((224⋅196)s/2ΓC(s)6L(s)Λ(2−s)
Λ(s)=(=((224⋅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 |
| 19 | 1+18T+144T2+737T3+144pT4+18p2T5+p3T6 |
good | 3 | 1+p2T+5p2T2+161T3+50p2T4+38p3T5+1945T6+38p4T7+50p4T8+161p3T9+5p6T10+p7T11+p6T12 |
| 5 | 1+6T+18T2+9pT3+81T4+87T5+109T6+87pT7+81p2T8+9p4T9+18p4T10+6p5T11+p6T12 |
| 7 | 1+6T+6T2+6T3+24pT4+384T5+191T6+384pT7+24p3T8+6p3T9+6p4T10+6p5T11+p6T12 |
| 11 | 1+6T−6T2−14T3+504T4+768T5−3409T6+768pT7+504p2T8−14p3T9−6p4T10+6p5T11+p6T12 |
| 13 | 1+15T+96T2+298T3−9pT4−6255T5−32679T6−6255pT7−9p3T8+298p3T9+96p4T10+15p5T11+p6T12 |
| 17 | 1+9T+36T2+40T3−369T4−2673T5−10927T6−2673pT7−369p2T8+40p3T9+36p4T10+9p5T11+p6T12 |
| 23 | 1+18T+180T2+1354T3+8784T4+49464T5+249209T6+49464pT7+8784p2T8+1354p3T9+180p4T10+18p5T11+p6T12 |
| 29 | 1+3T+36T2+378T3+1872T4+9921T5+94159T6+9921pT7+1872p2T8+378p3T9+36p4T10+3p5T11+p6T12 |
| 31 | 1−3T−60T2+59T3+2223T4+774T5−78969T6+774pT7+2223p2T8+59p3T9−60p4T10−3p5T11+p6T12 |
| 37 | (1+12T+150T2+907T3+150pT4+12p2T5+p3T6)2 |
| 41 | 1+15T+150T2+1000T3+8025T4+61965T5+472529T6+61965pT7+8025p2T8+1000p3T9+150p4T10+15p5T11+p6T12 |
| 43 | 1−21T+174T2−454T3−3879T4+58707T5−460263T6+58707pT7−3879p2T8−454p3T9+174p4T10−21p5T11+p6T12 |
| 47 | 1+21T+186T2+908T3+2715T4+657T5−54025T6+657pT7+2715p2T8+908p3T9+186p4T10+21p5T11+p6T12 |
| 53 | 1+27T+324T2+2178T3+11259T4+87453T5+753949T6+87453pT7+11259p2T8+2178p3T9+324p4T10+27p5T11+p6T12 |
| 59 | 1−6T+6T2+332T3−2208T4−10908T5+340589T6−10908pT7−2208p2T8+332p3T9+6p4T10−6p5T11+p6T12 |
| 61 | 1−12T−24T2+1238T3−7956T4−48996T5+978639T6−48996pT7−7956p2T8+1238p3T9−24p4T10−12p5T11+p6T12 |
| 67 | 1+18T+252T2+1838T3+4968T4−64692T5−990987T6−64692pT7+4968p2T8+1838p3T9+252p4T10+18p5T11+p6T12 |
| 71 | 1+18T+252T2+1814T3+5832T4−54324T5−887815T6−54324pT7+5832p2T8+1814p3T9+252p4T10+18p5T11+p6T12 |
| 73 | 1−36T+576T2−5760T3+51984T4−544320T5+5272343T6−544320pT7+51984p2T8−5760p3T9+576p4T10−36p5T11+p6T12 |
| 79 | 1+27T+180T2−1964T3−32490T4−11547T5+1937037T6−11547pT7−32490p2T8−1964p3T9+180p4T10+27p5T11+p6T12 |
| 83 | 1+12T−150T2−562T3+37494T4+98250T5−2840605T6+98250pT7+37494p2T8−562p3T9−150p4T10+12p5T11+p6T12 |
| 89 | 1+12T+54T2+1035T3−279T4−69891T5+3961T6−69891pT7−279p2T8+1035p3T9+54p4T10+12p5T11+p6T12 |
| 97 | 1+18T+270T2+4178T3+50940T4+535752T5+5820555T6+535752pT7+50940p2T8+4178p3T9+270p4T10+18p5T11+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
−7.00820039279161874905501137167, −6.49971060897362979511561371440, −6.38134130913356182863179587099, −6.34543112826713143912387064344, −6.28060821305869572482434432607, −6.21765520560200267778015873485, −5.99103586538089416765680574736, −5.54597359423337706076743522041, −5.39594318982216479586036883501, −5.19977406470742622286100624982, −5.19040623208105583996420403525, −5.15433618110883852317273453322, −4.84450292400711764204908889079, −4.43746854256803047399051037185, −4.29341750782648339354552986395, −4.23696712306679260547063002140, −4.20644719544482949991517078395, −3.97297030795175168094658397943, −3.43834989136773901717483753778, −3.25857511079660822708767588263, −2.84908086618182205097568939456, −2.68618014787347857699935654435, −2.24339989875866482874873369859, −2.03519893992654500254372132537, −1.95867123665135618429143645799, 0, 0, 0, 0, 0, 0,
1.95867123665135618429143645799, 2.03519893992654500254372132537, 2.24339989875866482874873369859, 2.68618014787347857699935654435, 2.84908086618182205097568939456, 3.25857511079660822708767588263, 3.43834989136773901717483753778, 3.97297030795175168094658397943, 4.20644719544482949991517078395, 4.23696712306679260547063002140, 4.29341750782648339354552986395, 4.43746854256803047399051037185, 4.84450292400711764204908889079, 5.15433618110883852317273453322, 5.19040623208105583996420403525, 5.19977406470742622286100624982, 5.39594318982216479586036883501, 5.54597359423337706076743522041, 5.99103586538089416765680574736, 6.21765520560200267778015873485, 6.28060821305869572482434432607, 6.34543112826713143912387064344, 6.38134130913356182863179587099, 6.49971060897362979511561371440, 7.00820039279161874905501137167
Plot not available for L-functions of degree greater than 10.