L(s) = 1 | − 133.·2-s + 64.3·3-s + 9.58e3·4-s + 1.95e4·5-s − 8.57e3·6-s + 1.77e5·7-s − 1.85e5·8-s − 1.59e6·9-s − 2.60e6·10-s − 3.36e5·11-s + 6.16e5·12-s + 1.41e7·13-s − 2.36e7·14-s + 1.25e6·15-s − 5.37e7·16-s + 1.10e8·17-s + 2.12e8·18-s − 1.18e8·19-s + 1.87e8·20-s + 1.14e7·21-s + 4.47e7·22-s − 7.03e8·23-s − 1.19e7·24-s − 8.37e8·25-s − 1.89e9·26-s − 2.04e8·27-s + 1.69e9·28-s + ⋯ |
L(s) = 1 | − 1.47·2-s + 0.0509·3-s + 1.16·4-s + 0.559·5-s − 0.0750·6-s + 0.569·7-s − 0.250·8-s − 0.997·9-s − 0.824·10-s − 0.0571·11-s + 0.0596·12-s + 0.815·13-s − 0.839·14-s + 0.0285·15-s − 0.801·16-s + 1.11·17-s + 1.46·18-s − 0.575·19-s + 0.654·20-s + 0.0290·21-s + 0.0842·22-s − 0.990·23-s − 0.0127·24-s − 0.686·25-s − 1.20·26-s − 0.101·27-s + 0.666·28-s + ⋯ |
Λ(s)=(=(197s/2ΓC(s)L(s)Λ(14−s)
Λ(s)=(=(197s/2ΓC(s+13/2)L(s)Λ(1−s)
Particular Values
L(7) |
≈ |
0.9840262625 |
L(21) |
≈ |
0.9840262625 |
L(215) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 197 | 1−5.84e13T |
good | 2 | 1+133.T+8.19e3T2 |
| 3 | 1−64.3T+1.59e6T2 |
| 5 | 1−1.95e4T+1.22e9T2 |
| 7 | 1−1.77e5T+9.68e10T2 |
| 11 | 1+3.36e5T+3.45e13T2 |
| 13 | 1−1.41e7T+3.02e14T2 |
| 17 | 1−1.10e8T+9.90e15T2 |
| 19 | 1+1.18e8T+4.20e16T2 |
| 23 | 1+7.03e8T+5.04e17T2 |
| 29 | 1+3.44e9T+1.02e19T2 |
| 31 | 1+5.07e9T+2.44e19T2 |
| 37 | 1−2.95e10T+2.43e20T2 |
| 41 | 1−3.95e10T+9.25e20T2 |
| 43 | 1+5.83e10T+1.71e21T2 |
| 47 | 1−8.53e10T+5.46e21T2 |
| 53 | 1+2.30e11T+2.60e22T2 |
| 59 | 1−2.95e11T+1.04e23T2 |
| 61 | 1−3.76e11T+1.61e23T2 |
| 67 | 1−5.34e11T+5.48e23T2 |
| 71 | 1−1.73e12T+1.16e24T2 |
| 73 | 1−6.32e11T+1.67e24T2 |
| 79 | 1+1.64e12T+4.66e24T2 |
| 83 | 1+7.58e11T+8.87e24T2 |
| 89 | 1−1.01e11T+2.19e25T2 |
| 97 | 1+1.06e12T+6.73e25T2 |
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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
−9.917110897775969962423120305841, −9.228999269754609107886690237397, −8.212401890454571390395868178109, −7.75244428763011171391723583953, −6.28122600901542719184391858681, −5.45246981919792835667509007961, −3.85512020974786597001394527392, −2.36883831769383456422097539829, −1.57310229908007807017715488488, −0.52716100692339385437479282583,
0.52716100692339385437479282583, 1.57310229908007807017715488488, 2.36883831769383456422097539829, 3.85512020974786597001394527392, 5.45246981919792835667509007961, 6.28122600901542719184391858681, 7.75244428763011171391723583953, 8.212401890454571390395868178109, 9.228999269754609107886690237397, 9.917110897775969962423120305841