L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s − 2·13-s − 2·15-s + 6·17-s + 4·19-s − 25-s + 27-s + 2·29-s − 4·33-s − 6·37-s − 2·39-s − 2·41-s + 4·43-s − 2·45-s + 6·51-s − 6·53-s + 8·55-s + 4·57-s + 12·59-s − 2·61-s + 4·65-s − 4·67-s + 6·73-s − 75-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.696·33-s − 0.986·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s + 0.840·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.56·59-s − 0.256·61-s + 0.496·65-s − 0.488·67-s + 0.702·73-s − 0.115·75-s + ⋯ |
Λ(s)=(=(9408s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(9408s/2ΓC(s+1/2)L(s)−Λ(1−s)
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1−T |
| 7 | 1 |
good | 5 | 1+2T+pT2 |
| 11 | 1+4T+pT2 |
| 13 | 1+2T+pT2 |
| 17 | 1−6T+pT2 |
| 19 | 1−4T+pT2 |
| 23 | 1+pT2 |
| 29 | 1−2T+pT2 |
| 31 | 1+pT2 |
| 37 | 1+6T+pT2 |
| 41 | 1+2T+pT2 |
| 43 | 1−4T+pT2 |
| 47 | 1+pT2 |
| 53 | 1+6T+pT2 |
| 59 | 1−12T+pT2 |
| 61 | 1+2T+pT2 |
| 67 | 1+4T+pT2 |
| 71 | 1+pT2 |
| 73 | 1−6T+pT2 |
| 79 | 1+16T+pT2 |
| 83 | 1+12T+pT2 |
| 89 | 1−14T+pT2 |
| 97 | 1+18T+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
−7.41162979563327424538391535036, −7.08463284614449031614302730659, −5.84366683169447599172191319438, −5.25233061805071063226919136648, −4.56778712711941505527081631821, −3.63417394911967134530507000276, −3.13295449004887072368969635069, −2.36822825472434546359232655114, −1.19651242282215311581594916621, 0,
1.19651242282215311581594916621, 2.36822825472434546359232655114, 3.13295449004887072368969635069, 3.63417394911967134530507000276, 4.56778712711941505527081631821, 5.25233061805071063226919136648, 5.84366683169447599172191319438, 7.08463284614449031614302730659, 7.41162979563327424538391535036