L(s) = 1 | + 3.82·2-s + 3·3-s + 6.65·4-s + 11.4·6-s + 7·7-s − 5.14·8-s + 9·9-s + 48.5·11-s + 19.9·12-s + 43.6·13-s + 26.7·14-s − 72.9·16-s + 67.6·17-s + 34.4·18-s − 93.2·19-s + 21·21-s + 185.·22-s + 104.·23-s − 15.4·24-s + 167.·26-s + 27·27-s + 46.5·28-s − 58.7·29-s − 9.08·31-s − 238.·32-s + 145.·33-s + 259.·34-s + ⋯ |
L(s) = 1 | + 1.35·2-s + 0.577·3-s + 0.832·4-s + 0.781·6-s + 0.377·7-s − 0.227·8-s + 0.333·9-s + 1.33·11-s + 0.480·12-s + 0.931·13-s + 0.511·14-s − 1.13·16-s + 0.965·17-s + 0.451·18-s − 1.12·19-s + 0.218·21-s + 1.80·22-s + 0.944·23-s − 0.131·24-s + 1.26·26-s + 0.192·27-s + 0.314·28-s − 0.376·29-s − 0.0526·31-s − 1.31·32-s + 0.768·33-s + 1.30·34-s + ⋯ |
Λ(s)=(=(525s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(525s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
5.521893815 |
L(21) |
≈ |
5.521893815 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1−3T |
| 5 | 1 |
| 7 | 1−7T |
good | 2 | 1−3.82T+8T2 |
| 11 | 1−48.5T+1.33e3T2 |
| 13 | 1−43.6T+2.19e3T2 |
| 17 | 1−67.6T+4.91e3T2 |
| 19 | 1+93.2T+6.85e3T2 |
| 23 | 1−104.T+1.21e4T2 |
| 29 | 1+58.7T+2.43e4T2 |
| 31 | 1+9.08T+2.97e4T2 |
| 37 | 1−252.T+5.06e4T2 |
| 41 | 1−276.T+6.89e4T2 |
| 43 | 1−92.6T+7.95e4T2 |
| 47 | 1−582.T+1.03e5T2 |
| 53 | 1+623.T+1.48e5T2 |
| 59 | 1+524.T+2.05e5T2 |
| 61 | 1+352.T+2.26e5T2 |
| 67 | 1−736.T+3.00e5T2 |
| 71 | 1+492.T+3.57e5T2 |
| 73 | 1+1.16e3T+3.89e5T2 |
| 79 | 1+872.T+4.93e5T2 |
| 83 | 1−529.T+5.71e5T2 |
| 89 | 1+385.T+7.04e5T2 |
| 97 | 1−463.T+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
−10.76709716702433480999084431519, −9.336058746616116421305830704442, −8.793176351752138474110594018678, −7.61419417613129886177638587302, −6.47579138051612609557328574560, −5.76664371588232783169228167220, −4.45370029574047547099551712284, −3.85831518143065839884360286752, −2.80628483670701080579322390267, −1.33890609301519855356138672872,
1.33890609301519855356138672872, 2.80628483670701080579322390267, 3.85831518143065839884360286752, 4.45370029574047547099551712284, 5.76664371588232783169228167220, 6.47579138051612609557328574560, 7.61419417613129886177638587302, 8.793176351752138474110594018678, 9.336058746616116421305830704442, 10.76709716702433480999084431519