| L(s) = 1 | − 2.33·2-s + 2.58·3-s + 3.45·4-s − 4.30·5-s − 6.04·6-s + 1.79·7-s − 3.40·8-s + 3.69·9-s + 10.0·10-s − 4.87·11-s + 8.94·12-s + 2.64·13-s − 4.19·14-s − 11.1·15-s + 1.03·16-s + 17-s − 8.62·18-s − 3.80·19-s − 14.8·20-s + 4.65·21-s + 11.3·22-s + 7.68·23-s − 8.79·24-s + 13.5·25-s − 6.18·26-s + 1.79·27-s + 6.21·28-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.49·3-s + 1.72·4-s − 1.92·5-s − 2.46·6-s + 0.679·7-s − 1.20·8-s + 1.23·9-s + 3.17·10-s − 1.46·11-s + 2.58·12-s + 0.734·13-s − 1.12·14-s − 2.87·15-s + 0.257·16-s + 0.242·17-s − 2.03·18-s − 0.872·19-s − 3.32·20-s + 1.01·21-s + 2.42·22-s + 1.60·23-s − 1.79·24-s + 2.70·25-s − 1.21·26-s + 0.345·27-s + 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 3 | \( 1 - 2.58T + 3T^{2} \) |
| 5 | \( 1 + 4.30T + 5T^{2} \) |
| 7 | \( 1 - 1.79T + 7T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 19 | \( 1 + 3.80T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 + 8.84T + 29T^{2} \) |
| 31 | \( 1 + 5.05T + 31T^{2} \) |
| 37 | \( 1 + 12.1T + 37T^{2} \) |
| 41 | \( 1 + 5.93T + 41T^{2} \) |
| 43 | \( 1 - 4.61T + 43T^{2} \) |
| 47 | \( 1 + 6.07T + 47T^{2} \) |
| 53 | \( 1 - 5.69T + 53T^{2} \) |
| 61 | \( 1 + 8.74T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 - 0.691T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 4.80T + 83T^{2} \) |
| 89 | \( 1 - 7.29T + 89T^{2} \) |
| 97 | \( 1 + 0.107T + 97T^{2} \) |
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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
−8.983303801550423223553882926322, −8.700047659181793753490411359763, −8.044233446589967973486894773265, −7.56271943682859209074167723905, −7.05161668551668992547205317543, −5.00254663202633215848973242646, −3.76158259099049763096083187268, −2.95357829517511742644497605647, −1.69646327185045402303115284470, 0,
1.69646327185045402303115284470, 2.95357829517511742644497605647, 3.76158259099049763096083187268, 5.00254663202633215848973242646, 7.05161668551668992547205317543, 7.56271943682859209074167723905, 8.044233446589967973486894773265, 8.700047659181793753490411359763, 8.983303801550423223553882926322
