L(s) = 1 | + 3-s − 4·7-s + 9-s − 4·13-s + 8·19-s − 4·21-s + 4·23-s + 27-s − 6·29-s + 8·31-s + 4·37-s − 4·39-s + 6·41-s + 4·43-s − 4·47-s + 9·49-s + 12·53-s + 8·57-s − 6·61-s − 4·63-s + 12·67-s + 4·69-s + 16·71-s + 8·79-s + 81-s − 12·83-s − 6·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.10·13-s + 1.83·19-s − 0.872·21-s + 0.834·23-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.657·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s + 1.64·53-s + 1.05·57-s − 0.768·61-s − 0.503·63-s + 1.46·67-s + 0.481·69-s + 1.89·71-s + 0.900·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + ⋯ |
Λ(s)=(=(2400s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(2400s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
1.773016203 |
L(21) |
≈ |
1.773016203 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1−T |
| 5 | 1 |
good | 7 | 1+4T+pT2 |
| 11 | 1+pT2 |
| 13 | 1+4T+pT2 |
| 17 | 1+pT2 |
| 19 | 1−8T+pT2 |
| 23 | 1−4T+pT2 |
| 29 | 1+6T+pT2 |
| 31 | 1−8T+pT2 |
| 37 | 1−4T+pT2 |
| 41 | 1−6T+pT2 |
| 43 | 1−4T+pT2 |
| 47 | 1+4T+pT2 |
| 53 | 1−12T+pT2 |
| 59 | 1+pT2 |
| 61 | 1+6T+pT2 |
| 67 | 1−12T+pT2 |
| 71 | 1−16T+pT2 |
| 73 | 1+pT2 |
| 79 | 1−8T+pT2 |
| 83 | 1+12T+pT2 |
| 89 | 1+10T+pT2 |
| 97 | 1+8T+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
−9.294439362155582311396720712775, −8.146688029147244337134377106094, −7.33074642876103566465615298847, −6.86435994056926016287184004578, −5.86257513923387218027991820847, −5.02892022270217661724574337422, −3.92685848464838036707491107183, −3.07713744288718387943470887456, −2.48768118964892092450282845760, −0.819834028977665625201711882906,
0.819834028977665625201711882906, 2.48768118964892092450282845760, 3.07713744288718387943470887456, 3.92685848464838036707491107183, 5.02892022270217661724574337422, 5.86257513923387218027991820847, 6.86435994056926016287184004578, 7.33074642876103566465615298847, 8.146688029147244337134377106094, 9.294439362155582311396720712775