| L(s) = 1 | − 2.44·2-s − 2.68·3-s + 3.99·4-s − 1.41·5-s + 6.58·6-s − 0.890·7-s − 4.87·8-s + 4.23·9-s + 3.46·10-s + 5.63·11-s − 10.7·12-s + 6.51·13-s + 2.17·14-s + 3.81·15-s + 3.94·16-s + 0.509·17-s − 10.3·18-s + 2.70·19-s − 5.65·20-s + 2.39·21-s − 13.7·22-s + 0.817·23-s + 13.1·24-s − 2.99·25-s − 15.9·26-s − 3.32·27-s − 3.55·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 1.55·3-s + 1.99·4-s − 0.633·5-s + 2.68·6-s − 0.336·7-s − 1.72·8-s + 1.41·9-s + 1.09·10-s + 1.69·11-s − 3.09·12-s + 1.80·13-s + 0.582·14-s + 0.983·15-s + 0.985·16-s + 0.123·17-s − 2.44·18-s + 0.619·19-s − 1.26·20-s + 0.522·21-s − 2.93·22-s + 0.170·23-s + 2.67·24-s − 0.598·25-s − 3.12·26-s − 0.639·27-s − 0.671·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 6047 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 + 2.68T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 0.890T + 7T^{2} \) |
| 11 | \( 1 - 5.63T + 11T^{2} \) |
| 13 | \( 1 - 6.51T + 13T^{2} \) |
| 17 | \( 1 - 0.509T + 17T^{2} \) |
| 19 | \( 1 - 2.70T + 19T^{2} \) |
| 23 | \( 1 - 0.817T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 - 6.30T + 31T^{2} \) |
| 37 | \( 1 + 9.26T + 37T^{2} \) |
| 41 | \( 1 + 0.0960T + 41T^{2} \) |
| 43 | \( 1 + 1.68T + 43T^{2} \) |
| 47 | \( 1 + 9.10T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 59 | \( 1 + 6.44T + 59T^{2} \) |
| 61 | \( 1 + 7.59T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 3.10T + 73T^{2} \) |
| 79 | \( 1 + 5.94T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 - 7.64T + 89T^{2} \) |
| 97 | \( 1 - 6.26T + 97T^{2} \) |
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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
−7.86834778069544268523183243863, −6.83019950206401462989510907991, −6.47429138534549565682833565322, −6.14508176235812418040281454185, −4.96450572029889671060010980011, −4.00043075484851184254873587698, −3.19101431190142515761383645742, −1.40579558544903093022366945003, −1.13024119369489997735000770598, 0,
1.13024119369489997735000770598, 1.40579558544903093022366945003, 3.19101431190142515761383645742, 4.00043075484851184254873587698, 4.96450572029889671060010980011, 6.14508176235812418040281454185, 6.47429138534549565682833565322, 6.83019950206401462989510907991, 7.86834778069544268523183243863
