L(s) = 1 | − 1.08·2-s + 9·3-s − 30.8·4-s − 9.73·6-s + 139.·7-s + 67.9·8-s + 81·9-s − 121·11-s − 277.·12-s + 646.·13-s − 150.·14-s + 913.·16-s + 1.37e3·17-s − 87.5·18-s + 1.90e3·19-s + 1.25e3·21-s + 130.·22-s − 343.·23-s + 611.·24-s − 698.·26-s + 729·27-s − 4.30e3·28-s + 53.5·29-s + 634.·31-s − 3.16e3·32-s − 1.08e3·33-s − 1.49e3·34-s + ⋯ |
L(s) = 1 | − 0.191·2-s + 0.577·3-s − 0.963·4-s − 0.110·6-s + 1.07·7-s + 0.375·8-s + 0.333·9-s − 0.301·11-s − 0.556·12-s + 1.06·13-s − 0.205·14-s + 0.891·16-s + 1.15·17-s − 0.0637·18-s + 1.21·19-s + 0.621·21-s + 0.0576·22-s − 0.135·23-s + 0.216·24-s − 0.202·26-s + 0.192·27-s − 1.03·28-s + 0.0118·29-s + 0.118·31-s − 0.545·32-s − 0.174·33-s − 0.221·34-s + ⋯ |
Λ(s)=(=(825s/2ΓC(s)L(s)Λ(6−s)
Λ(s)=(=(825s/2ΓC(s+5/2)L(s)Λ(1−s)
Particular Values
L(3) |
≈ |
2.781540747 |
L(21) |
≈ |
2.781540747 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1−9T |
| 5 | 1 |
| 11 | 1+121T |
good | 2 | 1+1.08T+32T2 |
| 7 | 1−139.T+1.68e4T2 |
| 13 | 1−646.T+3.71e5T2 |
| 17 | 1−1.37e3T+1.41e6T2 |
| 19 | 1−1.90e3T+2.47e6T2 |
| 23 | 1+343.T+6.43e6T2 |
| 29 | 1−53.5T+2.05e7T2 |
| 31 | 1−634.T+2.86e7T2 |
| 37 | 1+1.16e4T+6.93e7T2 |
| 41 | 1−1.88e4T+1.15e8T2 |
| 43 | 1+1.33e4T+1.47e8T2 |
| 47 | 1−2.25e3T+2.29e8T2 |
| 53 | 1−8.90e3T+4.18e8T2 |
| 59 | 1+1.12e4T+7.14e8T2 |
| 61 | 1−1.18e4T+8.44e8T2 |
| 67 | 1−5.73e4T+1.35e9T2 |
| 71 | 1−3.10e4T+1.80e9T2 |
| 73 | 1+5.60e4T+2.07e9T2 |
| 79 | 1+883.T+3.07e9T2 |
| 83 | 1+9.39e4T+3.93e9T2 |
| 89 | 1−2.17e4T+5.58e9T2 |
| 97 | 1−1.58e5T+8.58e9T2 |
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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
−9.404342941132456143639012171175, −8.482215120775309968013891966663, −8.057116307075131438285682697028, −7.25510354805086355741050604983, −5.70828310327807472909894539110, −5.03701218988820549792891851670, −3.99961804413794600935408905284, −3.16457992086396753472234850663, −1.62034891384160577330339635658, −0.843343357606116739670221804741,
0.843343357606116739670221804741, 1.62034891384160577330339635658, 3.16457992086396753472234850663, 3.99961804413794600935408905284, 5.03701218988820549792891851670, 5.70828310327807472909894539110, 7.25510354805086355741050604983, 8.057116307075131438285682697028, 8.482215120775309968013891966663, 9.404342941132456143639012171175