| L(s) = 1 | + 0.569·2-s + 3-s − 1.67·4-s + 1.10·5-s + 0.569·6-s + 2.67·7-s − 2.09·8-s + 9-s + 0.627·10-s − 5.58·11-s − 1.67·12-s + 5.66·13-s + 1.52·14-s + 1.10·15-s + 2.16·16-s − 17-s + 0.569·18-s − 8.17·19-s − 1.84·20-s + 2.67·21-s − 3.17·22-s + 1.82·23-s − 2.09·24-s − 3.78·25-s + 3.22·26-s + 27-s − 4.47·28-s + ⋯ |
| L(s) = 1 | + 0.402·2-s + 0.577·3-s − 0.837·4-s + 0.492·5-s + 0.232·6-s + 1.00·7-s − 0.739·8-s + 0.333·9-s + 0.198·10-s − 1.68·11-s − 0.483·12-s + 1.57·13-s + 0.406·14-s + 0.284·15-s + 0.540·16-s − 0.242·17-s + 0.134·18-s − 1.87·19-s − 0.412·20-s + 0.582·21-s − 0.677·22-s + 0.380·23-s − 0.427·24-s − 0.757·25-s + 0.632·26-s + 0.192·27-s − 0.845·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 0.569T + 2T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 + 5.58T + 11T^{2} \) |
| 13 | \( 1 - 5.66T + 13T^{2} \) |
| 19 | \( 1 + 8.17T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 + 0.370T + 29T^{2} \) |
| 31 | \( 1 + 0.983T + 31T^{2} \) |
| 37 | \( 1 + 7.61T + 37T^{2} \) |
| 41 | \( 1 + 0.900T + 41T^{2} \) |
| 43 | \( 1 + 0.382T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 6.33T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 - 4.73T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 3.19T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 + 6.69T + 83T^{2} \) |
| 89 | \( 1 + 5.70T + 89T^{2} \) |
| 97 | \( 1 + 6.95T + 97T^{2} \) |
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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
−7.79411921888586295371154847725, −6.73002342348887571545139910548, −5.81062443363066957166057680410, −5.43296051914952334856288787494, −4.53738376140752472925340186535, −4.07693095236051476199066589441, −3.15080084715088678667499973148, −2.27413020728568630688304868695, −1.47005380843801800770496788214, 0,
1.47005380843801800770496788214, 2.27413020728568630688304868695, 3.15080084715088678667499973148, 4.07693095236051476199066589441, 4.53738376140752472925340186535, 5.43296051914952334856288787494, 5.81062443363066957166057680410, 6.73002342348887571545139910548, 7.79411921888586295371154847725
