L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s + 3·11-s + 4·13-s + 16-s − 2·19-s + 3·20-s + 3·22-s + 6·23-s + 4·25-s + 4·26-s − 6·29-s − 5·31-s + 32-s + 2·37-s − 2·38-s + 3·40-s − 6·41-s − 10·43-s + 3·44-s + 6·46-s + 6·47-s + 4·50-s + 4·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s + 0.904·11-s + 1.10·13-s + 1/4·16-s − 0.458·19-s + 0.670·20-s + 0.639·22-s + 1.25·23-s + 4/5·25-s + 0.784·26-s − 1.11·29-s − 0.898·31-s + 0.176·32-s + 0.328·37-s − 0.324·38-s + 0.474·40-s − 0.937·41-s − 1.52·43-s + 0.452·44-s + 0.884·46-s + 0.875·47-s + 0.565·50-s + 0.554·52-s + ⋯ |
Λ(s)=(=(2646s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(2646s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
4.133596812 |
L(21) |
≈ |
4.133596812 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1−T |
| 3 | 1 |
| 7 | 1 |
good | 5 | 1−3T+pT2 |
| 11 | 1−3T+pT2 |
| 13 | 1−4T+pT2 |
| 17 | 1+pT2 |
| 19 | 1+2T+pT2 |
| 23 | 1−6T+pT2 |
| 29 | 1+6T+pT2 |
| 31 | 1+5T+pT2 |
| 37 | 1−2T+pT2 |
| 41 | 1+6T+pT2 |
| 43 | 1+10T+pT2 |
| 47 | 1−6T+pT2 |
| 53 | 1+9T+pT2 |
| 59 | 1−12T+pT2 |
| 61 | 1+8T+pT2 |
| 67 | 1−14T+pT2 |
| 71 | 1+pT2 |
| 73 | 1−7T+pT2 |
| 79 | 1−8T+pT2 |
| 83 | 1+3T+pT2 |
| 89 | 1+18T+pT2 |
| 97 | 1−T+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
−9.022301181159976542679631550139, −8.153752740732516759811873175243, −6.90390849159694968636551446179, −6.51972259556560194206176883369, −5.69537931970611061608929469001, −5.14293667469015758909535382312, −4.01126047180402972127592124525, −3.26671376711039705407751493003, −2.07276435338156744028852711093, −1.33346620954041525748246641205,
1.33346620954041525748246641205, 2.07276435338156744028852711093, 3.26671376711039705407751493003, 4.01126047180402972127592124525, 5.14293667469015758909535382312, 5.69537931970611061608929469001, 6.51972259556560194206176883369, 6.90390849159694968636551446179, 8.153752740732516759811873175243, 9.022301181159976542679631550139