| L(s) = 1 | − 0.277·2-s − 1.24·3-s − 7.92·4-s − 5·5-s + 0.345·6-s − 34.3·7-s + 4.41·8-s − 25.4·9-s + 1.38·10-s + 11·11-s + 9.85·12-s + 62.9·13-s + 9.52·14-s + 6.22·15-s + 62.1·16-s + 72.5·17-s + 7.06·18-s − 19·19-s + 39.6·20-s + 42.7·21-s − 3.05·22-s + 42.2·23-s − 5.49·24-s + 25·25-s − 17.4·26-s + 65.2·27-s + 272.·28-s + ⋯ |
| L(s) = 1 | − 0.0980·2-s − 0.239·3-s − 0.990·4-s − 0.447·5-s + 0.0234·6-s − 1.85·7-s + 0.195·8-s − 0.942·9-s + 0.0438·10-s + 0.301·11-s + 0.237·12-s + 1.34·13-s + 0.181·14-s + 0.107·15-s + 0.971·16-s + 1.03·17-s + 0.0924·18-s − 0.229·19-s + 0.442·20-s + 0.444·21-s − 0.0295·22-s + 0.382·23-s − 0.0467·24-s + 0.200·25-s − 0.131·26-s + 0.465·27-s + 1.83·28-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 | \( 1 + 0.277T + 8T^{2} \) |
| 3 | \( 1 + 1.24T + 27T^{2} \) |
| 7 | \( 1 + 34.3T + 343T^{2} \) |
| 13 | \( 1 - 62.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 42.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 165.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 28.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 129.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 280.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 79.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 441.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 173.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 351.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 331.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 705.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 24.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 209.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 768.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 906.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 338.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 824.T + 9.12e5T^{2} \) |
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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
−8.995590066073649015963570886165, −8.599917829935823278945434497826, −7.52762849433229005175261401276, −6.28052963838203376518458682430, −5.93577657556321326860102456920, −4.69911962071362272046334743198, −3.48045372470732664488661780296, −3.19550353072336862315726090801, −0.951859634980970987498103917020, 0,
0.951859634980970987498103917020, 3.19550353072336862315726090801, 3.48045372470732664488661780296, 4.69911962071362272046334743198, 5.93577657556321326860102456920, 6.28052963838203376518458682430, 7.52762849433229005175261401276, 8.599917829935823278945434497826, 8.995590066073649015963570886165
