| L(s) = 1 | − 256·2-s + 1.20e4·3-s + 6.55e4·4-s − 3.90e5·5-s − 3.08e6·6-s − 9.53e6·7-s − 1.67e7·8-s + 1.57e7·9-s + 1.00e8·10-s + 4.01e8·11-s + 7.88e8·12-s + 8.56e8·13-s + 2.44e9·14-s − 4.70e9·15-s + 4.29e9·16-s − 3.89e10·17-s − 4.03e9·18-s − 1.13e11·19-s − 2.56e10·20-s − 1.14e11·21-s − 1.02e11·22-s + 1.64e10·23-s − 2.01e11·24-s + 1.52e11·25-s − 2.19e11·26-s − 1.36e12·27-s − 6.24e11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.05·3-s + 0.5·4-s − 0.447·5-s − 0.749·6-s − 0.625·7-s − 0.353·8-s + 0.122·9-s + 0.316·10-s + 0.564·11-s + 0.529·12-s + 0.291·13-s + 0.442·14-s − 0.473·15-s + 0.250·16-s − 1.35·17-s − 0.0863·18-s − 1.53·19-s − 0.223·20-s − 0.662·21-s − 0.399·22-s + 0.0438·23-s − 0.374·24-s + 0.200·25-s − 0.205·26-s − 0.929·27-s − 0.312·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 256T \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 3 | \( 1 - 1.20e4T + 1.29e8T^{2} \) |
| 7 | \( 1 + 9.53e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 4.01e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 8.56e8T + 8.65e18T^{2} \) |
| 17 | \( 1 + 3.89e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 1.13e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.64e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.27e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 1.63e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.75e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 2.95e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.37e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.65e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 7.25e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.62e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.46e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.03e14T + 1.10e31T^{2} \) |
| 71 | \( 1 - 9.39e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.54e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 8.30e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 6.14e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 4.67e15T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.01e17T + 5.95e33T^{2} \) |
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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
−15.80511956302180615323028788421, −14.63122790578675903841159792684, −12.97191703121270810045472509464, −11.09736117497202661028797722908, −9.293081749459928340961998145999, −8.330234956749635530572762170744, −6.65212381802034175078729787725, −3.73470910704275074307727234376, −2.16006779529385751767201227750, 0,
2.16006779529385751767201227750, 3.73470910704275074307727234376, 6.65212381802034175078729787725, 8.330234956749635530572762170744, 9.293081749459928340961998145999, 11.09736117497202661028797722908, 12.97191703121270810045472509464, 14.63122790578675903841159792684, 15.80511956302180615323028788421
