L(s) = 1 | − 6·2-s + 3-s + 23·4-s + 6·5-s − 6·6-s + 3·7-s − 72·8-s − 2·9-s − 36·10-s − 2·11-s + 23·12-s + 6·13-s − 18·14-s + 6·15-s + 199·16-s − 3·17-s + 12·18-s + 5·19-s + 138·20-s + 3·21-s + 12·22-s − 22·23-s − 72·24-s + 31·25-s − 36·26-s + 3·27-s + 69·28-s + ⋯ |
L(s) = 1 | − 4.24·2-s + 0.577·3-s + 23/2·4-s + 2.68·5-s − 2.44·6-s + 1.13·7-s − 25.4·8-s − 2/3·9-s − 11.3·10-s − 0.603·11-s + 6.63·12-s + 1.66·13-s − 4.81·14-s + 1.54·15-s + 49.7·16-s − 0.727·17-s + 2.82·18-s + 1.14·19-s + 30.8·20-s + 0.654·21-s + 2.55·22-s − 4.58·23-s − 14.6·24-s + 31/5·25-s − 7.06·26-s + 0.577·27-s + 13.0·28-s + ⋯ |
Λ(s)=(=((3116)s/2ΓC(s)8L(s)Λ(2−s)
Λ(s)=(=((3116)s/2ΓC(s+1/2)8L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
0.6268554449 |
L(21) |
≈ |
0.6268554449 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 31 | 1 |
good | 2 | (1+3T+pT2+T4+p3T6+3p3T7+p4T8)2 |
| 3 | 1−T+pT2−8T3+8T4+7T5+2pT6+56T7−137T8+56pT9+2p3T10+7p3T11+8p4T12−8p5T13+p7T14−p7T15+p8T16 |
| 5 | (1−3T−2T2−3T3+51T4−3pT5−2p2T6−3p3T7+p4T8)2 |
| 7 | 1−3T+pT2−36T3+108T4−219T5+122pT6−2628T7+5483T8−2628pT9+122p3T10−219p3T11+108p4T12−36p5T13+p7T14−3p7T15+p8T16 |
| 11 | 1+2T−9T2−42T3−124T4−84T5+629T6+3734T7+13467T8+3734pT9+629p2T10−84p3T11−124p4T12−42p5T13−9p6T14+2p7T15+p8T16 |
| 13 | 1−6T+pT2+102T3−792T4+2892T5−1009T6−36864T7+214223T8−36864pT9−1009p2T10+2892p3T11−792p4T12+102p5T13+p7T14−6p7T15+p8T16 |
| 17 | 1+3T+7T2−144T3−732T4−1641T5+1934T6+49698T7+169253T8+49698pT9+1934p2T10−1641p3T11−732p4T12−144p5T13+7p6T14+3p7T15+p8T16 |
| 19 | 1−5T+pT2−160T3+800T4−2765T5+914pT6−88480T7+312079T8−88480pT9+914p3T10−2765p3T11+800p4T12−160p5T13+p7T14−5p7T15+p8T16 |
| 23 | (1+11T+38T2+125T3+821T4+125pT5+38p2T6+11p3T7+p4T8)2 |
| 29 | (1+5T+31T2+115T3+96T4+115pT5+31p2T6+5p3T7+p4T8)2 |
| 37 | (1+4T−57T2−4T3+3368T4−4pT5−57p2T6+4p3T7+p4T8)2 |
| 41 | 1+8T+pT2−528T3−6224T4−39696T5−2521T6+1579436T7+15159727T8+1579436pT9−2521p2T10−39696p3T11−6224p4T12−528p5T13+p7T14+8p7T15+p8T16 |
| 43 | 1−T+28T2−443T3−232T4+6152T5−2224T6+797146T7−5551967T8+797146pT9−2224p2T10+6152p3T11−232p4T12−443p5T13+28p6T14−p7T15+p8T16 |
| 47 | (1−7T−23T2+385T3−1284T4+385pT5−23p2T6−7p3T7+p4T8)2 |
| 53 | 1−21T+323T2−3588T3+35568T4−324633T5+2767366T6−22572954T7+167761313T8−22572954pT9+2767366p2T10−324633p3T11+35568p4T12−3588p5T13+323p6T14−21p7T15+p8T16 |
| 59 | 1+5T−T2−900T3−8060T4−17835T5+134126T6+3072320T7+18108079T8+3072320pT9+134126p2T10−17835p3T11−8060p4T12−900p5T13−p6T14+5p7T15+p8T16 |
| 61 | (1+4T+46T2+4pT3+p2T4)4 |
| 67 | (1−4T−117T2+4T3+12128T4+4pT5−117p2T6−4p3T7+p4T8)2 |
| 71 | 1−7T−4T2+1227T3−13364T4+49044T5+214904T6−6815734T7+62113357T8−6815734pT9+214904p2T10+49044p3T11−13364p4T12+1227p5T13−4p6T14−7p7T15+p8T16 |
| 73 | 1−21T+208T2+297T3−27162T4+338082T5−1041544T6−15918264T7+260450783T8−15918264pT9−1041544p2T10+338082p3T11−27162p4T12+297p5T13+208p6T14−21p7T15+p8T16 |
| 79 | 1+pT2−p3T6−p4T8−p5T10+p7T14+p8T16 |
| 83 | 1+14T+183T2+42T3−7462T4−179508T5+16211T6+9573326T7+219347043T8+9573326pT9+16211p2T10−179508p3T11−7462p4T12+42p5T13+183p6T14+14p7T15+p8T16 |
| 89 | (1+5T−29T2−785T3−624T4−785pT5−29p2T6+5p3T7+p4T8)2 |
| 97 | (1+3T+182T2−345T3+16591T4−345pT5+182p2T6+3p3T7+p4T8)2 |
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L(s)=p∏ j=1∏16(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−4.11785362386263796915429844500, −4.04649440355080535665346968830, −3.91171049420904226129017720215, −3.85065884087830684620774896299, −3.79902916918042833384525830623, −3.42586768077915414588102727396, −3.26575178780436515427297702479, −3.08577623125226845634543434428, −2.94233324374835259721826749716, −2.82856616796589576643485553010, −2.79916642327298704340249926732, −2.76964943195737707604289556942, −2.45464289867226235506060072542, −2.38639132884310982688608931021, −2.11634963730662732143897465048, −1.96202862526523815842587911030, −1.83563558880710970582614596099, −1.73153472587432438911458436904, −1.66163765365748164876243263975, −1.60007125143123260745248295434, −1.28973116896153040280460046712, −1.08008172402387320784562462080, −0.71539967031560814626829494069, −0.68018641171872669942947275026, −0.15859220531520854148455475806,
0.15859220531520854148455475806, 0.68018641171872669942947275026, 0.71539967031560814626829494069, 1.08008172402387320784562462080, 1.28973116896153040280460046712, 1.60007125143123260745248295434, 1.66163765365748164876243263975, 1.73153472587432438911458436904, 1.83563558880710970582614596099, 1.96202862526523815842587911030, 2.11634963730662732143897465048, 2.38639132884310982688608931021, 2.45464289867226235506060072542, 2.76964943195737707604289556942, 2.79916642327298704340249926732, 2.82856616796589576643485553010, 2.94233324374835259721826749716, 3.08577623125226845634543434428, 3.26575178780436515427297702479, 3.42586768077915414588102727396, 3.79902916918042833384525830623, 3.85065884087830684620774896299, 3.91171049420904226129017720215, 4.04649440355080535665346968830, 4.11785362386263796915429844500
Plot not available for L-functions of degree greater than 10.