L(s) = 1 | + 5·5-s + 22·7-s − 12·11-s + 38·13-s + 105·17-s + 157·19-s − 117·23-s + 25·25-s − 66·29-s + 25·31-s + 110·35-s + 314·37-s + 504·41-s − 380·43-s − 252·47-s + 141·49-s − 3·53-s − 60·55-s − 318·59-s + 293·61-s + 190·65-s + 322·67-s − 120·71-s + 44·73-s − 264·77-s − 917·79-s + 309·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.18·7-s − 0.328·11-s + 0.810·13-s + 1.49·17-s + 1.89·19-s − 1.06·23-s + 1/5·25-s − 0.422·29-s + 0.144·31-s + 0.531·35-s + 1.39·37-s + 1.91·41-s − 1.34·43-s − 0.782·47-s + 0.411·49-s − 0.00777·53-s − 0.147·55-s − 0.701·59-s + 0.614·61-s + 0.362·65-s + 0.587·67-s − 0.200·71-s + 0.0705·73-s − 0.390·77-s − 1.30·79-s + 0.408·83-s + ⋯ |
Λ(s)=(=(2160s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(2160s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
3.535899116 |
L(21) |
≈ |
3.535899116 |
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 | 1−22T+p3T2 |
| 11 | 1+12T+p3T2 |
| 13 | 1−38T+p3T2 |
| 17 | 1−105T+p3T2 |
| 19 | 1−157T+p3T2 |
| 23 | 1+117T+p3T2 |
| 29 | 1+66T+p3T2 |
| 31 | 1−25T+p3T2 |
| 37 | 1−314T+p3T2 |
| 41 | 1−504T+p3T2 |
| 43 | 1+380T+p3T2 |
| 47 | 1+252T+p3T2 |
| 53 | 1+3T+p3T2 |
| 59 | 1+318T+p3T2 |
| 61 | 1−293T+p3T2 |
| 67 | 1−322T+p3T2 |
| 71 | 1+120T+p3T2 |
| 73 | 1−44T+p3T2 |
| 79 | 1+917T+p3T2 |
| 83 | 1−309T+p3T2 |
| 89 | 1+1272T+p3T2 |
| 97 | 1−1328T+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
−8.600224127229223504736382379721, −7.82717285630202922929664470569, −7.50239371313490625592441508240, −6.14223266970085526524793455680, −5.55887653795517989118154070915, −4.86109668270508664550614353224, −3.78582226702215965501330683334, −2.84500460654417303921612111355, −1.63496497663678019117224170964, −0.944946556762391867027290063342,
0.944946556762391867027290063342, 1.63496497663678019117224170964, 2.84500460654417303921612111355, 3.78582226702215965501330683334, 4.86109668270508664550614353224, 5.55887653795517989118154070915, 6.14223266970085526524793455680, 7.50239371313490625592441508240, 7.82717285630202922929664470569, 8.600224127229223504736382379721