L(s) = 1 | + 3.23·2-s + 2.30·3-s + 2.49·4-s + 5·5-s + 7.48·6-s − 5.43·7-s − 17.8·8-s − 21.6·9-s + 16.1·10-s + 11·11-s + 5.75·12-s + 31.6·13-s − 17.5·14-s + 11.5·15-s − 77.7·16-s − 13.7·17-s − 70.1·18-s + 19·19-s + 12.4·20-s − 12.5·21-s + 35.6·22-s − 11.5·23-s − 41.2·24-s + 25·25-s + 102.·26-s − 112.·27-s − 13.5·28-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.444·3-s + 0.311·4-s + 0.447·5-s + 0.509·6-s − 0.293·7-s − 0.788·8-s − 0.802·9-s + 0.512·10-s + 0.301·11-s + 0.138·12-s + 0.675·13-s − 0.335·14-s + 0.198·15-s − 1.21·16-s − 0.195·17-s − 0.918·18-s + 0.229·19-s + 0.139·20-s − 0.130·21-s + 0.345·22-s − 0.104·23-s − 0.350·24-s + 0.200·25-s + 0.773·26-s − 0.801·27-s − 0.0913·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 - 3.23T + 8T^{2} \) |
| 3 | \( 1 - 2.30T + 27T^{2} \) |
| 7 | \( 1 + 5.43T + 343T^{2} \) |
| 13 | \( 1 - 31.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 13.7T + 4.91e3T^{2} \) |
| 23 | \( 1 + 11.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 211.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 267.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 271.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 20.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 296.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 311.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 399.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 139.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 66.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 219.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 8.47T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 206.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 26.1T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.150324978105838675393406780489, −8.440333482840756861567075097538, −7.29660786138187265543068459726, −6.12144344545718912059327388833, −5.77344600199141063004655327433, −4.70735677470620728729000460175, −3.64039365213386350558831429637, −3.04582507180906790021039567924, −1.84174494716756956305066249145, 0,
1.84174494716756956305066249145, 3.04582507180906790021039567924, 3.64039365213386350558831429637, 4.70735677470620728729000460175, 5.77344600199141063004655327433, 6.12144344545718912059327388833, 7.29660786138187265543068459726, 8.440333482840756861567075097538, 9.150324978105838675393406780489