| L(s) = 1 | − 2-s − 1.01·3-s + 4-s − 1.91·5-s + 1.01·6-s + 4.92·7-s − 8-s − 1.97·9-s + 1.91·10-s − 0.589·11-s − 1.01·12-s + 2.74·13-s − 4.92·14-s + 1.93·15-s + 16-s − 1.63·17-s + 1.97·18-s − 1.23·19-s − 1.91·20-s − 4.97·21-s + 0.589·22-s + 1.68·23-s + 1.01·24-s − 1.32·25-s − 2.74·26-s + 5.03·27-s + 4.92·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.583·3-s + 0.5·4-s − 0.857·5-s + 0.412·6-s + 1.86·7-s − 0.353·8-s − 0.659·9-s + 0.606·10-s − 0.177·11-s − 0.291·12-s + 0.760·13-s − 1.31·14-s + 0.500·15-s + 0.250·16-s − 0.395·17-s + 0.466·18-s − 0.284·19-s − 0.428·20-s − 1.08·21-s + 0.125·22-s + 0.350·23-s + 0.206·24-s − 0.264·25-s − 0.537·26-s + 0.968·27-s + 0.930·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 + 1.01T + 3T^{2} \) |
| 5 | \( 1 + 1.91T + 5T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 0.589T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 + 1.63T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 - 1.68T + 23T^{2} \) |
| 29 | \( 1 + 7.46T + 29T^{2} \) |
| 31 | \( 1 - 1.54T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 + 9.29T + 43T^{2} \) |
| 47 | \( 1 + 3.13T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 7.96T + 61T^{2} \) |
| 67 | \( 1 - 2.77T + 67T^{2} \) |
| 71 | \( 1 - 2.80T + 71T^{2} \) |
| 73 | \( 1 - 0.952T + 73T^{2} \) |
| 79 | \( 1 + 9.00T + 79T^{2} \) |
| 83 | \( 1 - 3.06T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 8.87T + 97T^{2} \) |
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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
−7.958683203006362786361998515558, −7.77183689664662842375784687063, −6.69570770732408390795265798053, −5.87709958248005721392046086456, −5.09769595011872196424989861655, −4.40452855877381410550394123205, −3.41802720457098998303406235015, −2.17897338324948972598599307894, −1.24289943189213891202065801965, 0,
1.24289943189213891202065801965, 2.17897338324948972598599307894, 3.41802720457098998303406235015, 4.40452855877381410550394123205, 5.09769595011872196424989861655, 5.87709958248005721392046086456, 6.69570770732408390795265798053, 7.77183689664662842375784687063, 7.958683203006362786361998515558
