L(s) = 1 | + 5·5-s + 34·7-s + 16·11-s + 58·13-s + 70·17-s − 4·19-s − 134·23-s + 25·25-s + 242·29-s − 100·31-s + 170·35-s − 438·37-s + 138·41-s − 178·43-s + 22·47-s + 813·49-s − 162·53-s + 80·55-s − 268·59-s + 250·61-s + 290·65-s − 422·67-s − 852·71-s + 306·73-s + 544·77-s + 456·79-s + 434·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.83·7-s + 0.438·11-s + 1.23·13-s + 0.998·17-s − 0.0482·19-s − 1.21·23-s + 1/5·25-s + 1.54·29-s − 0.579·31-s + 0.821·35-s − 1.94·37-s + 0.525·41-s − 0.631·43-s + 0.0682·47-s + 2.37·49-s − 0.419·53-s + 0.196·55-s − 0.591·59-s + 0.524·61-s + 0.553·65-s − 0.769·67-s − 1.42·71-s + 0.490·73-s + 0.805·77-s + 0.649·79-s + 0.573·83-s + ⋯ |
Λ(s)=(=(720s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(720s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
3.171541832 |
L(21) |
≈ |
3.171541832 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1−pT |
good | 7 | 1−34T+p3T2 |
| 11 | 1−16T+p3T2 |
| 13 | 1−58T+p3T2 |
| 17 | 1−70T+p3T2 |
| 19 | 1+4T+p3T2 |
| 23 | 1+134T+p3T2 |
| 29 | 1−242T+p3T2 |
| 31 | 1+100T+p3T2 |
| 37 | 1+438T+p3T2 |
| 41 | 1−138T+p3T2 |
| 43 | 1+178T+p3T2 |
| 47 | 1−22T+p3T2 |
| 53 | 1+162T+p3T2 |
| 59 | 1+268T+p3T2 |
| 61 | 1−250T+p3T2 |
| 67 | 1+422T+p3T2 |
| 71 | 1+12pT+p3T2 |
| 73 | 1−306T+p3T2 |
| 79 | 1−456T+p3T2 |
| 83 | 1−434T+p3T2 |
| 89 | 1−726T+p3T2 |
| 97 | 1−1378T+p3T2 |
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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
−10.22572480046913745011930392958, −8.947607958223266007249640291711, −8.335377338324811771435674064641, −7.59054497170985846597906126568, −6.36464635179123227994521505831, −5.49226044370723787902006471980, −4.60805128938907848460513995844, −3.52565889202379224030494240673, −1.92193744145234397781099933324, −1.16728557410045569590268749643,
1.16728557410045569590268749643, 1.92193744145234397781099933324, 3.52565889202379224030494240673, 4.60805128938907848460513995844, 5.49226044370723787902006471980, 6.36464635179123227994521505831, 7.59054497170985846597906126568, 8.335377338324811771435674064641, 8.947607958223266007249640291711, 10.22572480046913745011930392958