| L(s) = 1 | − 0.949·2-s + 1.30·3-s − 1.09·4-s − 1.23·6-s + 1.82·7-s + 2.94·8-s − 1.30·9-s + 3.24·11-s − 1.42·12-s + 1.15·13-s − 1.73·14-s − 0.596·16-s − 3.53·17-s + 1.24·18-s − 2.07·19-s + 2.37·21-s − 3.07·22-s + 2.93·23-s + 3.82·24-s − 1.09·26-s − 5.60·27-s − 2.00·28-s − 5.23·29-s − 10.0·31-s − 5.31·32-s + 4.21·33-s + 3.35·34-s + ⋯ |
| L(s) = 1 | − 0.671·2-s + 0.750·3-s − 0.549·4-s − 0.504·6-s + 0.691·7-s + 1.04·8-s − 0.436·9-s + 0.977·11-s − 0.412·12-s + 0.320·13-s − 0.464·14-s − 0.149·16-s − 0.856·17-s + 0.293·18-s − 0.476·19-s + 0.518·21-s − 0.656·22-s + 0.612·23-s + 0.780·24-s − 0.215·26-s − 1.07·27-s − 0.379·28-s − 0.972·29-s − 1.79·31-s − 0.940·32-s + 0.734·33-s + 0.575·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 0.949T + 2T^{2} \) |
| 3 | \( 1 - 1.30T + 3T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 + 3.53T + 17T^{2} \) |
| 19 | \( 1 + 2.07T + 19T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 + 5.23T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 + 8.57T + 53T^{2} \) |
| 59 | \( 1 - 3.47T + 59T^{2} \) |
| 61 | \( 1 + 1.91T + 61T^{2} \) |
| 67 | \( 1 - 1.85T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 0.875T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 + 2.08T + 83T^{2} \) |
| 89 | \( 1 - 6.90T + 89T^{2} \) |
| 97 | \( 1 + 8.32T + 97T^{2} \) |
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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
−8.183826695247824301430980530389, −7.43776229751824787481869474325, −6.67570888845430898188827482689, −5.66522309139040204204242296509, −4.85677318767974979936990592958, −4.03636015946896460209135731808, −3.41941659965215272570252974901, −2.12182768425154856188706450323, −1.43556047293116941897070793564, 0,
1.43556047293116941897070793564, 2.12182768425154856188706450323, 3.41941659965215272570252974901, 4.03636015946896460209135731808, 4.85677318767974979936990592958, 5.66522309139040204204242296509, 6.67570888845430898188827482689, 7.43776229751824787481869474325, 8.183826695247824301430980530389
