| L(s) = 1 | + 2-s − 1.27·3-s + 4-s + 1.77·5-s − 1.27·6-s + 1.20·7-s + 8-s − 1.37·9-s + 1.77·10-s − 2.07·11-s − 1.27·12-s + 0.395·13-s + 1.20·14-s − 2.26·15-s + 16-s − 1.89·17-s − 1.37·18-s + 1.68·19-s + 1.77·20-s − 1.53·21-s − 2.07·22-s − 6.28·23-s − 1.27·24-s − 1.85·25-s + 0.395·26-s + 5.57·27-s + 1.20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.736·3-s + 0.5·4-s + 0.792·5-s − 0.520·6-s + 0.456·7-s + 0.353·8-s − 0.458·9-s + 0.560·10-s − 0.625·11-s − 0.368·12-s + 0.109·13-s + 0.322·14-s − 0.583·15-s + 0.250·16-s − 0.460·17-s − 0.324·18-s + 0.387·19-s + 0.396·20-s − 0.335·21-s − 0.442·22-s − 1.31·23-s − 0.260·24-s − 0.371·25-s + 0.0776·26-s + 1.07·27-s + 0.228·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 0.395T + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 1.68T + 19T^{2} \) |
| 23 | \( 1 + 6.28T + 23T^{2} \) |
| 29 | \( 1 + 1.81T + 29T^{2} \) |
| 37 | \( 1 - 3.16T + 37T^{2} \) |
| 41 | \( 1 + 7.96T + 41T^{2} \) |
| 43 | \( 1 + 0.353T + 43T^{2} \) |
| 47 | \( 1 + 9.44T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 8.70T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 - 0.388T + 67T^{2} \) |
| 71 | \( 1 + 1.64T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 7.32T + 79T^{2} \) |
| 83 | \( 1 - 5.71T + 83T^{2} \) |
| 89 | \( 1 - 7.55T + 89T^{2} \) |
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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
−7.69275962819348143364815056059, −6.68775905796603437270358414747, −6.11804525309603307810508156740, −5.55203257483328423578143641628, −5.03091826347808272620250390786, −4.25774237444129352809164786276, −3.21587596703775283105740470585, −2.33252549241358574359862152291, −1.53591716470845027649528147779, 0,
1.53591716470845027649528147779, 2.33252549241358574359862152291, 3.21587596703775283105740470585, 4.25774237444129352809164786276, 5.03091826347808272620250390786, 5.55203257483328423578143641628, 6.11804525309603307810508156740, 6.68775905796603437270358414747, 7.69275962819348143364815056059
