L(s) = 1 | − 2.51·2-s − 3-s + 4.34·4-s + 2.51·6-s − 0.173·7-s − 5.90·8-s + 9-s + 5.08·11-s − 4.34·12-s − 1.82·13-s + 0.436·14-s + 6.19·16-s + 4.24·17-s − 2.51·18-s − 8.62·19-s + 0.173·21-s − 12.8·22-s − 3.16·23-s + 5.90·24-s + 4.60·26-s − 27-s − 0.753·28-s + 29-s + 3.22·31-s − 3.78·32-s − 5.08·33-s − 10.6·34-s + ⋯ |
L(s) = 1 | − 1.78·2-s − 0.577·3-s + 2.17·4-s + 1.02·6-s − 0.0655·7-s − 2.08·8-s + 0.333·9-s + 1.53·11-s − 1.25·12-s − 0.506·13-s + 0.116·14-s + 1.54·16-s + 1.02·17-s − 0.593·18-s − 1.97·19-s + 0.0378·21-s − 2.72·22-s − 0.659·23-s + 1.20·24-s + 0.902·26-s − 0.192·27-s − 0.142·28-s + 0.185·29-s + 0.578·31-s − 0.669·32-s − 0.884·33-s − 1.83·34-s + ⋯ |
Λ(s)=(=(2175s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(2175s/2ΓC(s+1/2)L(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 | 3 | 1+T |
| 5 | 1 |
| 29 | 1−T |
good | 2 | 1+2.51T+2T2 |
| 7 | 1+0.173T+7T2 |
| 11 | 1−5.08T+11T2 |
| 13 | 1+1.82T+13T2 |
| 17 | 1−4.24T+17T2 |
| 19 | 1+8.62T+19T2 |
| 23 | 1+3.16T+23T2 |
| 31 | 1−3.22T+31T2 |
| 37 | 1+1.97T+37T2 |
| 41 | 1+9.96T+41T2 |
| 43 | 1+7.91T+43T2 |
| 47 | 1−8.66T+47T2 |
| 53 | 1+5.40T+53T2 |
| 59 | 1−7.66T+59T2 |
| 61 | 1−2.76T+61T2 |
| 67 | 1+8.13T+67T2 |
| 71 | 1+6.25T+71T2 |
| 73 | 1−6.74T+73T2 |
| 79 | 1−4.54T+79T2 |
| 83 | 1−11.6T+83T2 |
| 89 | 1+16.4T+89T2 |
| 97 | 1+2.74T+97T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−8.634289428123115981019371274873, −8.188114274881213956221648721078, −7.14982176866421071439925990887, −6.57881900002486994031333451663, −6.01355311778597859270131890359, −4.67338120914734568974097030334, −3.56343859013510770513136402419, −2.14254364341070777381892870074, −1.28165043048159175423304649322, 0,
1.28165043048159175423304649322, 2.14254364341070777381892870074, 3.56343859013510770513136402419, 4.67338120914734568974097030334, 6.01355311778597859270131890359, 6.57881900002486994031333451663, 7.14982176866421071439925990887, 8.188114274881213956221648721078, 8.634289428123115981019371274873