L(s) = 1 | − 3·3-s + 16.2·7-s + 9·9-s + 40.2·11-s + 19.7·13-s + 83.0·17-s + 48.8·19-s − 48.6·21-s − 1.61·23-s − 27·27-s − 24.5·29-s + 12.4·31-s − 120.·33-s + 325.·37-s − 59.3·39-s − 242.·41-s − 367.·43-s + 204.·47-s − 80.2·49-s − 249.·51-s − 61.5·53-s − 146.·57-s + 112.·59-s + 477.·61-s + 145.·63-s − 558.·67-s + 4.83·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.875·7-s + 0.333·9-s + 1.10·11-s + 0.422·13-s + 1.18·17-s + 0.589·19-s − 0.505·21-s − 0.0146·23-s − 0.192·27-s − 0.157·29-s + 0.0719·31-s − 0.636·33-s + 1.44·37-s − 0.243·39-s − 0.923·41-s − 1.30·43-s + 0.634·47-s − 0.233·49-s − 0.683·51-s − 0.159·53-s − 0.340·57-s + 0.247·59-s + 1.00·61-s + 0.291·63-s − 1.01·67-s + 0.00844·69-s + ⋯ |
Λ(s)=(=(1200s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(1200s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
2.369189929 |
L(21) |
≈ |
2.369189929 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+3T |
| 5 | 1 |
good | 7 | 1−16.2T+343T2 |
| 11 | 1−40.2T+1.33e3T2 |
| 13 | 1−19.7T+2.19e3T2 |
| 17 | 1−83.0T+4.91e3T2 |
| 19 | 1−48.8T+6.85e3T2 |
| 23 | 1+1.61T+1.21e4T2 |
| 29 | 1+24.5T+2.43e4T2 |
| 31 | 1−12.4T+2.97e4T2 |
| 37 | 1−325.T+5.06e4T2 |
| 41 | 1+242.T+6.89e4T2 |
| 43 | 1+367.T+7.95e4T2 |
| 47 | 1−204.T+1.03e5T2 |
| 53 | 1+61.5T+1.48e5T2 |
| 59 | 1−112.T+2.05e5T2 |
| 61 | 1−477.T+2.26e5T2 |
| 67 | 1+558.T+3.00e5T2 |
| 71 | 1+558.T+3.57e5T2 |
| 73 | 1−1.01e3T+3.89e5T2 |
| 79 | 1+1.15e3T+4.93e5T2 |
| 83 | 1−1.15e3T+5.71e5T2 |
| 89 | 1−96.9T+7.04e5T2 |
| 97 | 1+1.15e3T+9.12e5T2 |
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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
−9.478131657047385873086771160621, −8.489527145532970048176229211555, −7.73911056024936449867241792804, −6.84372110389886155498715598437, −5.95799084770097863095443770661, −5.17382705644969217853794916366, −4.25407457691096052783135073048, −3.28569540606277144488958352111, −1.69619500963758275343443964988, −0.892085465030633368401629993636,
0.892085465030633368401629993636, 1.69619500963758275343443964988, 3.28569540606277144488958352111, 4.25407457691096052783135073048, 5.17382705644969217853794916366, 5.95799084770097863095443770661, 6.84372110389886155498715598437, 7.73911056024936449867241792804, 8.489527145532970048176229211555, 9.478131657047385873086771160621