L(s) = 1 | + 2·5-s − 2·7-s − 4·11-s + 13-s − 2·19-s + 4·23-s − 25-s − 2·31-s − 4·35-s + 10·37-s − 2·41-s + 8·43-s − 3·49-s + 12·53-s − 8·55-s − 12·59-s + 6·61-s + 2·65-s − 6·67-s − 8·71-s − 2·73-s + 8·77-s − 12·79-s + 4·83-s − 14·89-s − 2·91-s − 4·95-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.20·11-s + 0.277·13-s − 0.458·19-s + 0.834·23-s − 1/5·25-s − 0.359·31-s − 0.676·35-s + 1.64·37-s − 0.312·41-s + 1.21·43-s − 3/7·49-s + 1.64·53-s − 1.07·55-s − 1.56·59-s + 0.768·61-s + 0.248·65-s − 0.733·67-s − 0.949·71-s − 0.234·73-s + 0.911·77-s − 1.35·79-s + 0.439·83-s − 1.48·89-s − 0.209·91-s − 0.410·95-s + ⋯ |
Λ(s)=(=(7488s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(7488s/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 |
| 13 | 1−T |
good | 5 | 1−2T+pT2 |
| 7 | 1+2T+pT2 |
| 11 | 1+4T+pT2 |
| 17 | 1+pT2 |
| 19 | 1+2T+pT2 |
| 23 | 1−4T+pT2 |
| 29 | 1+pT2 |
| 31 | 1+2T+pT2 |
| 37 | 1−10T+pT2 |
| 41 | 1+2T+pT2 |
| 43 | 1−8T+pT2 |
| 47 | 1+pT2 |
| 53 | 1−12T+pT2 |
| 59 | 1+12T+pT2 |
| 61 | 1−6T+pT2 |
| 67 | 1+6T+pT2 |
| 71 | 1+8T+pT2 |
| 73 | 1+2T+pT2 |
| 79 | 1+12T+pT2 |
| 83 | 1−4T+pT2 |
| 89 | 1+14T+pT2 |
| 97 | 1+10T+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
−7.49026915499148598420863229304, −6.82611641620904034974269009642, −5.94660273142565104367971439251, −5.69655062923021187304922056070, −4.77219915335120526458588257092, −3.93999038741216645545048796411, −2.87117150112182190841690637877, −2.45734264446474454889575501395, −1.30745574631339359324468269776, 0,
1.30745574631339359324468269776, 2.45734264446474454889575501395, 2.87117150112182190841690637877, 3.93999038741216645545048796411, 4.77219915335120526458588257092, 5.69655062923021187304922056070, 5.94660273142565104367971439251, 6.82611641620904034974269009642, 7.49026915499148598420863229304