L(s) = 1 | − 4.71·2-s + 27.6·3-s − 9.75·4-s − 79.0·5-s − 130.·6-s + 189.·7-s + 196.·8-s + 524.·9-s + 372.·10-s − 396.·11-s − 270.·12-s + 722.·13-s − 891.·14-s − 2.18e3·15-s − 616.·16-s − 684.·17-s − 2.47e3·18-s − 2.52e3·19-s + 770.·20-s + 5.23e3·21-s + 1.87e3·22-s + 3.87e3·23-s + 5.45e3·24-s + 3.11e3·25-s − 3.40e3·26-s + 7.79e3·27-s − 1.84e3·28-s + ⋯ |
L(s) = 1 | − 0.833·2-s + 1.77·3-s − 0.304·4-s − 1.41·5-s − 1.48·6-s + 1.45·7-s + 1.08·8-s + 2.15·9-s + 1.17·10-s − 0.989·11-s − 0.541·12-s + 1.18·13-s − 1.21·14-s − 2.51·15-s − 0.602·16-s − 0.574·17-s − 1.79·18-s − 1.60·19-s + 0.430·20-s + 2.59·21-s + 0.824·22-s + 1.52·23-s + 1.93·24-s + 0.998·25-s − 0.988·26-s + 2.05·27-s − 0.444·28-s + ⋯ |
Λ(s)=(=(197s/2ΓC(s)L(s)Λ(6−s)
Λ(s)=(=(197s/2ΓC(s+5/2)L(s)Λ(1−s)
Particular Values
L(3) |
≈ |
1.985663163 |
L(21) |
≈ |
1.985663163 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 197 | 1−3.88e4T |
good | 2 | 1+4.71T+32T2 |
| 3 | 1−27.6T+243T2 |
| 5 | 1+79.0T+3.12e3T2 |
| 7 | 1−189.T+1.68e4T2 |
| 11 | 1+396.T+1.61e5T2 |
| 13 | 1−722.T+3.71e5T2 |
| 17 | 1+684.T+1.41e6T2 |
| 19 | 1+2.52e3T+2.47e6T2 |
| 23 | 1−3.87e3T+6.43e6T2 |
| 29 | 1+2.25e3T+2.05e7T2 |
| 31 | 1−5.83e3T+2.86e7T2 |
| 37 | 1−1.58e4T+6.93e7T2 |
| 41 | 1+5.10e3T+1.15e8T2 |
| 43 | 1−1.36e4T+1.47e8T2 |
| 47 | 1−2.55e4T+2.29e8T2 |
| 53 | 1−2.89e3T+4.18e8T2 |
| 59 | 1+698.T+7.14e8T2 |
| 61 | 1−1.46e4T+8.44e8T2 |
| 67 | 1−6.47e4T+1.35e9T2 |
| 71 | 1+4.14e4T+1.80e9T2 |
| 73 | 1+4.97e3T+2.07e9T2 |
| 79 | 1+3.49e4T+3.07e9T2 |
| 83 | 1−2.79e4T+3.93e9T2 |
| 89 | 1−1.24e5T+5.58e9T2 |
| 97 | 1+1.35e5T+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
−11.17486610235758670720280296413, −10.62176192545037162551305041742, −9.077294852098335710042586748660, −8.387424369754199961657178971456, −8.083177695634992083728651414618, −7.30251388707580509119347301313, −4.58901825809281318883919487563, −3.98419717346803907051562394322, −2.39885886187902438878229633830, −0.965217301508788902871111023886,
0.965217301508788902871111023886, 2.39885886187902438878229633830, 3.98419717346803907051562394322, 4.58901825809281318883919487563, 7.30251388707580509119347301313, 8.083177695634992083728651414618, 8.387424369754199961657178971456, 9.077294852098335710042586748660, 10.62176192545037162551305041742, 11.17486610235758670720280296413