L(s) = 1 | − 0.585·3-s − 5-s + 4.82·7-s − 2.65·9-s − 2·11-s − 2.24·13-s + 0.585·15-s − 4.82·17-s − 19-s − 2.82·21-s + 6·23-s + 25-s + 3.31·27-s − 10.4·29-s + 1.17·31-s + 1.17·33-s − 4.82·35-s + 10.2·37-s + 1.31·39-s − 7.65·41-s − 0.828·43-s + 2.65·45-s − 0.828·47-s + 16.3·49-s + 2.82·51-s + 5.07·53-s + 2·55-s + ⋯ |
L(s) = 1 | − 0.338·3-s − 0.447·5-s + 1.82·7-s − 0.885·9-s − 0.603·11-s − 0.621·13-s + 0.151·15-s − 1.17·17-s − 0.229·19-s − 0.617·21-s + 1.25·23-s + 0.200·25-s + 0.637·27-s − 1.94·29-s + 0.210·31-s + 0.203·33-s − 0.816·35-s + 1.68·37-s + 0.210·39-s − 1.19·41-s − 0.126·43-s + 0.396·45-s − 0.120·47-s + 2.33·49-s + 0.396·51-s + 0.696·53-s + 0.269·55-s + ⋯ |
Λ(s)=(=(6080s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(6080s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
1.387196386 |
L(21) |
≈ |
1.387196386 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1+T |
| 19 | 1+T |
good | 3 | 1+0.585T+3T2 |
| 7 | 1−4.82T+7T2 |
| 11 | 1+2T+11T2 |
| 13 | 1+2.24T+13T2 |
| 17 | 1+4.82T+17T2 |
| 23 | 1−6T+23T2 |
| 29 | 1+10.4T+29T2 |
| 31 | 1−1.17T+31T2 |
| 37 | 1−10.2T+37T2 |
| 41 | 1+7.65T+41T2 |
| 43 | 1+0.828T+43T2 |
| 47 | 1+0.828T+47T2 |
| 53 | 1−5.07T+53T2 |
| 59 | 1−2.34T+59T2 |
| 61 | 1−2.34T+61T2 |
| 67 | 1+0.585T+67T2 |
| 71 | 1+10.8T+71T2 |
| 73 | 1−14.4T+73T2 |
| 79 | 1−9.65T+79T2 |
| 83 | 1−9.31T+83T2 |
| 89 | 1−10.4T+89T2 |
| 97 | 1+1.75T+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
−7.947861460022356608331365400666, −7.56816008764371744985267929475, −6.73293544784854002754262875083, −5.76684419830009614012899265730, −4.98504766596448602259716081332, −4.77616891203080074289430267124, −3.75088571733376059673956504636, −2.59335884053265231702517697252, −1.93929606032299414233200589226, −0.61401219380394009344543808522,
0.61401219380394009344543808522, 1.93929606032299414233200589226, 2.59335884053265231702517697252, 3.75088571733376059673956504636, 4.77616891203080074289430267124, 4.98504766596448602259716081332, 5.76684419830009614012899265730, 6.73293544784854002754262875083, 7.56816008764371744985267929475, 7.947861460022356608331365400666