L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s + 7-s − 8-s + 6·9-s − 5·11-s + 3·12-s + 6·13-s − 14-s + 16-s + 17-s − 6·18-s − 3·19-s + 3·21-s + 5·22-s − 3·24-s − 6·26-s + 9·27-s + 28-s − 6·29-s − 4·31-s − 32-s − 15·33-s − 34-s + 6·36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s + 0.377·7-s − 0.353·8-s + 2·9-s − 1.50·11-s + 0.866·12-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.41·18-s − 0.688·19-s + 0.654·21-s + 1.06·22-s − 0.612·24-s − 1.17·26-s + 1.73·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 2.61·33-s − 0.171·34-s + 36-s + ⋯ |
Λ(s)=(=(350s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(350s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
1.717087778 |
L(21) |
≈ |
1.717087778 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+T |
| 5 | 1 |
| 7 | 1−T |
good | 3 | 1−pT+pT2 |
| 11 | 1+5T+pT2 |
| 13 | 1−6T+pT2 |
| 17 | 1−T+pT2 |
| 19 | 1+3T+pT2 |
| 23 | 1+pT2 |
| 29 | 1+6T+pT2 |
| 31 | 1+4T+pT2 |
| 37 | 1+8T+pT2 |
| 41 | 1−11T+pT2 |
| 43 | 1−8T+pT2 |
| 47 | 1+2T+pT2 |
| 53 | 1+4T+pT2 |
| 59 | 1−4T+pT2 |
| 61 | 1+2T+pT2 |
| 67 | 1+9T+pT2 |
| 71 | 1+10T+pT2 |
| 73 | 1−7T+pT2 |
| 79 | 1+2T+pT2 |
| 83 | 1+11T+pT2 |
| 89 | 1+11T+pT2 |
| 97 | 1−10T+pT2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.04646116195134762631293439185, −10.47861198024909662647793577652, −9.334015251861622392606705723868, −8.632062123138897033456433282966, −7.982704647601194275162559588174, −7.28202966725007180028492050863, −5.74372284090584252527668413358, −4.02569848626692601267512154263, −2.90932005195183109143742073385, −1.76448136119712918970498775230,
1.76448136119712918970498775230, 2.90932005195183109143742073385, 4.02569848626692601267512154263, 5.74372284090584252527668413358, 7.28202966725007180028492050863, 7.982704647601194275162559588174, 8.632062123138897033456433282966, 9.334015251861622392606705723868, 10.47861198024909662647793577652, 11.04646116195134762631293439185