L(s) = 1 | + 2·5-s − 7-s + 2·11-s + 2·13-s − 2·17-s + 19-s − 6·23-s − 25-s + 8·31-s − 2·35-s + 6·37-s + 8·43-s + 6·47-s + 49-s + 4·55-s + 10·61-s + 4·65-s + 4·67-s − 8·71-s + 6·73-s − 2·77-s − 8·79-s − 6·83-s − 4·85-s + 8·89-s − 2·91-s + 2·95-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.603·11-s + 0.554·13-s − 0.485·17-s + 0.229·19-s − 1.25·23-s − 1/5·25-s + 1.43·31-s − 0.338·35-s + 0.986·37-s + 1.21·43-s + 0.875·47-s + 1/7·49-s + 0.539·55-s + 1.28·61-s + 0.496·65-s + 0.488·67-s − 0.949·71-s + 0.702·73-s − 0.227·77-s − 0.900·79-s − 0.658·83-s − 0.433·85-s + 0.847·89-s − 0.209·91-s + 0.205·95-s + ⋯ |
Λ(s)=(=(4788s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(4788s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
2.358631092 |
L(21) |
≈ |
2.358631092 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 7 | 1+T |
| 19 | 1−T |
good | 5 | 1−2T+pT2 |
| 11 | 1−2T+pT2 |
| 13 | 1−2T+pT2 |
| 17 | 1+2T+pT2 |
| 23 | 1+6T+pT2 |
| 29 | 1+pT2 |
| 31 | 1−8T+pT2 |
| 37 | 1−6T+pT2 |
| 41 | 1+pT2 |
| 43 | 1−8T+pT2 |
| 47 | 1−6T+pT2 |
| 53 | 1+pT2 |
| 59 | 1+pT2 |
| 61 | 1−10T+pT2 |
| 67 | 1−4T+pT2 |
| 71 | 1+8T+pT2 |
| 73 | 1−6T+pT2 |
| 79 | 1+8T+pT2 |
| 83 | 1+6T+pT2 |
| 89 | 1−8T+pT2 |
| 97 | 1−6T+pT2 |
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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
−8.369957576806470730042697469395, −7.55824816512863986169112032651, −6.63950471091123955223945426540, −6.09827076954569557762365258618, −5.61382453770497446170760864869, −4.45380207567394009332607936005, −3.85108594955125640162344451216, −2.74500890226059552917810012261, −1.96786711767249338800192239096, −0.866090107514273170957529972213,
0.866090107514273170957529972213, 1.96786711767249338800192239096, 2.74500890226059552917810012261, 3.85108594955125640162344451216, 4.45380207567394009332607936005, 5.61382453770497446170760864869, 6.09827076954569557762365258618, 6.63950471091123955223945426540, 7.55824816512863986169112032651, 8.369957576806470730042697469395