| L(s) = 1 | + 2.71·2-s + 2.09·3-s + 5.35·4-s + 5-s + 5.67·6-s − 7-s + 9.08·8-s + 1.37·9-s + 2.71·10-s + 11.1·12-s + 1.54·13-s − 2.71·14-s + 2.09·15-s + 13.9·16-s − 6.50·17-s + 3.73·18-s + 6.44·19-s + 5.35·20-s − 2.09·21-s − 4.80·23-s + 19.0·24-s + 25-s + 4.19·26-s − 3.39·27-s − 5.35·28-s + 4.72·29-s + 5.67·30-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 1.20·3-s + 2.67·4-s + 0.447·5-s + 2.31·6-s − 0.377·7-s + 3.21·8-s + 0.459·9-s + 0.857·10-s + 3.23·12-s + 0.428·13-s − 0.724·14-s + 0.540·15-s + 3.48·16-s − 1.57·17-s + 0.880·18-s + 1.47·19-s + 1.19·20-s − 0.456·21-s − 1.00·23-s + 3.88·24-s + 0.200·25-s + 0.822·26-s − 0.653·27-s − 1.01·28-s + 0.876·29-s + 1.03·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.72509735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.72509735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.71T + 2T^{2} \) |
| 3 | \( 1 - 2.09T + 3T^{2} \) |
| 13 | \( 1 - 1.54T + 13T^{2} \) |
| 17 | \( 1 + 6.50T + 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 + 4.80T + 23T^{2} \) |
| 29 | \( 1 - 4.72T + 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 + 5.91T + 37T^{2} \) |
| 41 | \( 1 + 2.19T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 0.998T + 47T^{2} \) |
| 53 | \( 1 + 8.13T + 53T^{2} \) |
| 59 | \( 1 - 6.77T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 5.75T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 - 6.90T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 6.07T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.266598536729604568274481377166, −7.36589767341792813265289270304, −6.81963053285168767070784771732, −5.98776050336196576118291233779, −5.42214522098646978274221176422, −4.42575890673416268327089557922, −3.79381207486506760598959989677, −3.04612839860769149089622498688, −2.45038228854887388475242272488, −1.64291954022611911909276919891,
1.64291954022611911909276919891, 2.45038228854887388475242272488, 3.04612839860769149089622498688, 3.79381207486506760598959989677, 4.42575890673416268327089557922, 5.42214522098646978274221176422, 5.98776050336196576118291233779, 6.81963053285168767070784771732, 7.36589767341792813265289270304, 8.266598536729604568274481377166
