| L(s) = 1 | − 3.14·2-s − 1.23·3-s + 1.87·4-s + 20.8·5-s + 3.88·6-s + 19.2·8-s − 25.4·9-s − 65.5·10-s − 25.4·11-s − 2.32·12-s + 43.9·13-s − 25.7·15-s − 75.5·16-s + 135.·17-s + 80.0·18-s − 7.85·19-s + 39.2·20-s + 79.8·22-s − 59.6·23-s − 23.7·24-s + 310.·25-s − 138.·26-s + 64.8·27-s − 87.5·29-s + 81.0·30-s − 54.0·31-s + 83.4·32-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 0.237·3-s + 0.234·4-s + 1.86·5-s + 0.264·6-s + 0.850·8-s − 0.943·9-s − 2.07·10-s − 0.696·11-s − 0.0558·12-s + 0.937·13-s − 0.443·15-s − 1.17·16-s + 1.93·17-s + 1.04·18-s − 0.0948·19-s + 0.438·20-s + 0.773·22-s − 0.541·23-s − 0.202·24-s + 2.48·25-s − 1.04·26-s + 0.462·27-s − 0.560·29-s + 0.493·30-s − 0.313·31-s + 0.460·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.492985623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.492985623\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 + 3.14T + 8T^{2} \) |
| 3 | \( 1 + 1.23T + 27T^{2} \) |
| 5 | \( 1 - 20.8T + 125T^{2} \) |
| 11 | \( 1 + 25.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 135.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 7.85T + 6.85e3T^{2} \) |
| 23 | \( 1 + 59.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 87.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 54.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 94.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 16.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 294.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 668.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 28.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 407.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 922.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 529.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 114.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 236.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 722.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 397.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 715.T + 9.12e5T^{2} \) |
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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
−8.687169651617614366515975946641, −8.213504567390597291999244807248, −7.23635354302848285842545888261, −6.22828390525284035882777518463, −5.55448910662537930709708128737, −5.15186716654336787452046112379, −3.55203752481138886926492201511, −2.43771177623557502449851981735, −1.56518768886477509364514382083, −0.69424834684879336231714725044,
0.69424834684879336231714725044, 1.56518768886477509364514382083, 2.43771177623557502449851981735, 3.55203752481138886926492201511, 5.15186716654336787452046112379, 5.55448910662537930709708128737, 6.22828390525284035882777518463, 7.23635354302848285842545888261, 8.213504567390597291999244807248, 8.687169651617614366515975946641
