L(s) = 1 | − 5.64·3-s − 12.1·5-s − 33.8·7-s + 4.91·9-s + 20.6·13-s + 68.7·15-s + 60.1·17-s + 105.·19-s + 191.·21-s + 186.·23-s + 23.2·25-s + 124.·27-s − 135.·29-s + 89.6·31-s + 412.·35-s − 183.·37-s − 116.·39-s − 372.·41-s − 294.·43-s − 59.8·45-s + 8.47·47-s + 802.·49-s − 339.·51-s − 219.·53-s − 598.·57-s + 260.·59-s + 514.·61-s + ⋯ |
L(s) = 1 | − 1.08·3-s − 1.08·5-s − 1.82·7-s + 0.181·9-s + 0.440·13-s + 1.18·15-s + 0.858·17-s + 1.27·19-s + 1.98·21-s + 1.69·23-s + 0.185·25-s + 0.889·27-s − 0.867·29-s + 0.519·31-s + 1.98·35-s − 0.817·37-s − 0.479·39-s − 1.41·41-s − 1.04·43-s − 0.198·45-s + 0.0263·47-s + 2.33·49-s − 0.933·51-s − 0.569·53-s − 1.39·57-s + 0.574·59-s + 1.08·61-s + ⋯ |
Λ(s)=(=(968s/2ΓC(s)L(s)−Λ(4−s)
Λ(s)=(=(968s/2ΓC(s+3/2)L(s)−Λ(1−s)
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 11 | 1 |
good | 3 | 1+5.64T+27T2 |
| 5 | 1+12.1T+125T2 |
| 7 | 1+33.8T+343T2 |
| 13 | 1−20.6T+2.19e3T2 |
| 17 | 1−60.1T+4.91e3T2 |
| 19 | 1−105.T+6.85e3T2 |
| 23 | 1−186.T+1.21e4T2 |
| 29 | 1+135.T+2.43e4T2 |
| 31 | 1−89.6T+2.97e4T2 |
| 37 | 1+183.T+5.06e4T2 |
| 41 | 1+372.T+6.89e4T2 |
| 43 | 1+294.T+7.95e4T2 |
| 47 | 1−8.47T+1.03e5T2 |
| 53 | 1+219.T+1.48e5T2 |
| 59 | 1−260.T+2.05e5T2 |
| 61 | 1−514.T+2.26e5T2 |
| 67 | 1+263.T+3.00e5T2 |
| 71 | 1+309.T+3.57e5T2 |
| 73 | 1−210.T+3.89e5T2 |
| 79 | 1−934.T+4.93e5T2 |
| 83 | 1−238.T+5.71e5T2 |
| 89 | 1+620.T+7.04e5T2 |
| 97 | 1−1.20e3T+9.12e5T2 |
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
−9.367698632416796455037964875748, −8.374630339502662723126232512963, −7.22098374619536424239544202892, −6.71203552661881784892388461265, −5.75257980302797775298357908967, −5.00273773875573917296600761384, −3.53735200960973284239642405761, −3.18748682264423219084776720327, −0.898603883452229434451006403324, 0,
0.898603883452229434451006403324, 3.18748682264423219084776720327, 3.53735200960973284239642405761, 5.00273773875573917296600761384, 5.75257980302797775298357908967, 6.71203552661881784892388461265, 7.22098374619536424239544202892, 8.374630339502662723126232512963, 9.367698632416796455037964875748