L(s) = 1 | − 4·3-s + 10·5-s + 2·7-s + 6·9-s − 2·11-s − 2·13-s − 40·15-s − 4·17-s + 3·19-s − 8·21-s − 14·23-s + 37·25-s − 5·27-s − 5·29-s + 14·31-s + 8·33-s + 20·35-s − 9·37-s + 8·39-s − 12·41-s − 18·43-s + 60·45-s − 3·47-s + 2·49-s + 16·51-s + 9·53-s − 20·55-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 4.47·5-s + 0.755·7-s + 2·9-s − 0.603·11-s − 0.554·13-s − 10.3·15-s − 0.970·17-s + 0.688·19-s − 1.74·21-s − 2.91·23-s + 37/5·25-s − 0.962·27-s − 0.928·29-s + 2.51·31-s + 1.39·33-s + 3.38·35-s − 1.47·37-s + 1.28·39-s − 1.87·41-s − 2.74·43-s + 8.94·45-s − 0.437·47-s + 2/7·49-s + 2.24·51-s + 1.23·53-s − 2.69·55-s + ⋯ |
Λ(s)=(=((224⋅312⋅76)s/2ΓC(s)6L(s)Λ(2−s)
Λ(s)=(=((224⋅312⋅76)s/2ΓC(s+1/2)6L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
0.8627257114 |
L(21) |
≈ |
0.8627257114 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+4T+10T2+7pT3+10pT4+4p2T5+p3T6 |
| 7 | 1−2T+2T2+19T3+2pT4−2p2T5+p3T6 |
good | 5 | (1−pT+19T2−47T3+19pT4−p3T5+p3T6)2 |
| 11 | (1+T+7T2+5pT3+7pT4+p2T5+p3T6)2 |
| 13 | 1+2T−32T2−2pT3+730T4+230T5−10729T6+230pT7+730p2T8−2p4T9−32p4T10+2p5T11+p6T12 |
| 17 | 1+4T+9T2+92T3+58T4−20T5+5393T6−20pT7+58p2T8+92p3T9+9p4T10+4p5T11+p6T12 |
| 19 | 1−3T−42T2+61T3+69pT4−726T5−27501T6−726pT7+69p3T8+61p3T9−42p4T10−3p5T11+p6T12 |
| 23 | (1+7T+73T2+319T3+73pT4+7p2T5+p3T6)2 |
| 29 | 1+5T−30T2−371T3−185T4+6020T5+44357T6+6020pT7−185p2T8−371p3T9−30p4T10+5p5T11+p6T12 |
| 31 | 1−14T+58T2−250T3+2992T4−9728T5−11857T6−9728pT7+2992p2T8−250p3T9+58p4T10−14p5T11+p6T12 |
| 37 | 1+9T−21T2−268T3+1293T4+4875T5−42882T6+4875pT7+1293p2T8−268p3T9−21p4T10+9p5T11+p6T12 |
| 41 | 1+12T−18T2−78T3+7470T4+24546T5−158105T6+24546pT7+7470p2T8−78p3T9−18p4T10+12p5T11+p6T12 |
| 43 | 1+18T+114T2+682T3+7188T4+33492T5+63039T6+33492pT7+7188p2T8+682p3T9+114p4T10+18p5T11+p6T12 |
| 47 | 1+3T−108T2−267T3+7263T4+9786T5−360137T6+9786pT7+7263p2T8−267p3T9−108p4T10+3p5T11+p6T12 |
| 53 | 1−9T−36T2+873T3−1179T4−26334T5+272077T6−26334pT7−1179p2T8+873p3T9−36p4T10−9p5T11+p6T12 |
| 59 | 1+4T−60T2−994T3−1304T4+464pT5+7381pT6+464p2T7−1304p2T8−994p3T9−60p4T10+4p5T11+p6T12 |
| 61 | 1−4T−32T2−650T3+292T4+19532T5+306323T6+19532pT7+292p2T8−650p3T9−32p4T10−4p5T11+p6T12 |
| 67 | 1+5T−118T2−327T3+8263T4+1138T5−609341T6+1138pT7+8263p2T8−327p3T9−118p4T10+5p5T11+p6T12 |
| 71 | (1+7T+163T2+895T3+163pT4+7p2T5+p3T6)2 |
| 73 | 1+25T+254T2+2073T3+20533T4+115046T5+366817T6+115046pT7+20533p2T8+2073p3T9+254p4T10+25p5T11+p6T12 |
| 79 | 1+7T−44T2−19T3−1043T4−28016T5+109223T6−28016pT7−1043p2T8−19p3T9−44p4T10+7p5T11+p6T12 |
| 83 | 1+8T−180T2−518T3+29404T4+32420T5−2713585T6+32420pT7+29404p2T8−518p3T9−180p4T10+8p5T11+p6T12 |
| 89 | 1+9T−180T2−729T3+31041T4+54846T5−2925911T6+54846pT7+31041p2T8−729p3T9−180p4T10+9p5T11+p6T12 |
| 97 | 1+28T+257T2+2820T3+59506T4+545924T5+3126001T6+545924pT7+59506p2T8+2820p3T9+257p4T10+28p5T11+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
−5.33180840133492559648434901668, −5.19607774219170389925403083351, −5.16061646378169559277793887487, −5.15905943986277968099351958805, −4.68701181304165230751247305788, −4.60723857661338138857017007134, −4.42936745977076228009183857697, −4.20748437566498736808595903149, −4.01518092750174708105673252263, −3.94543305387240282319054500557, −3.77606647391854451661155897704, −3.12480129779635462651186558067, −3.10029542898103028857316454742, −2.96648913038438996129268562835, −2.74724866109442533872490762208, −2.66538709856972608246273600499, −1.98945166766857644454963637492, −1.98521449318396799475856611423, −1.98406000552015288990483591268, −1.80679661295125280428201686411, −1.73673665905131937270034253148, −1.46888033376599572451699296229, −1.17688736407831848937463953416, −0.42949306127341024110573346703, −0.21725722712362141910799107253,
0.21725722712362141910799107253, 0.42949306127341024110573346703, 1.17688736407831848937463953416, 1.46888033376599572451699296229, 1.73673665905131937270034253148, 1.80679661295125280428201686411, 1.98406000552015288990483591268, 1.98521449318396799475856611423, 1.98945166766857644454963637492, 2.66538709856972608246273600499, 2.74724866109442533872490762208, 2.96648913038438996129268562835, 3.10029542898103028857316454742, 3.12480129779635462651186558067, 3.77606647391854451661155897704, 3.94543305387240282319054500557, 4.01518092750174708105673252263, 4.20748437566498736808595903149, 4.42936745977076228009183857697, 4.60723857661338138857017007134, 4.68701181304165230751247305788, 5.15905943986277968099351958805, 5.16061646378169559277793887487, 5.19607774219170389925403083351, 5.33180840133492559648434901668
Plot not available for L-functions of degree greater than 10.