| L(s) = 1 | + 2-s + 1.92·3-s + 4-s + 5-s + 1.92·6-s − 7-s + 8-s + 0.704·9-s + 10-s + 1.92·12-s − 5.24·13-s − 14-s + 1.92·15-s + 16-s − 4.30·17-s + 0.704·18-s − 0.740·19-s + 20-s − 1.92·21-s − 0.856·23-s + 1.92·24-s + 25-s − 5.24·26-s − 4.41·27-s − 28-s − 8.37·29-s + 1.92·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.11·3-s + 0.5·4-s + 0.447·5-s + 0.785·6-s − 0.377·7-s + 0.353·8-s + 0.234·9-s + 0.316·10-s + 0.555·12-s − 1.45·13-s − 0.267·14-s + 0.496·15-s + 0.250·16-s − 1.04·17-s + 0.166·18-s − 0.169·19-s + 0.223·20-s − 0.420·21-s − 0.178·23-s + 0.392·24-s + 0.200·25-s − 1.02·26-s − 0.850·27-s − 0.188·28-s − 1.55·29-s + 0.351·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.92T + 3T^{2} \) |
| 13 | \( 1 + 5.24T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 + 0.740T + 19T^{2} \) |
| 23 | \( 1 + 0.856T + 23T^{2} \) |
| 29 | \( 1 + 8.37T + 29T^{2} \) |
| 31 | \( 1 - 1.10T + 31T^{2} \) |
| 37 | \( 1 + 1.77T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 1.13T + 43T^{2} \) |
| 47 | \( 1 - 8.40T + 47T^{2} \) |
| 53 | \( 1 + 8.80T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 8.93T + 67T^{2} \) |
| 71 | \( 1 + 4.50T + 71T^{2} \) |
| 73 | \( 1 + 3.33T + 73T^{2} \) |
| 79 | \( 1 + 1.56T + 79T^{2} \) |
| 83 | \( 1 - 0.609T + 83T^{2} \) |
| 89 | \( 1 - 1.46T + 89T^{2} \) |
| 97 | \( 1 - 8.41T + 97T^{2} \) |
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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
−7.34793214951995574771418430818, −6.84749608249199482537514238955, −6.02055038284962576508075428512, −5.28819175944476233062883916550, −4.57345750909378303031907465411, −3.76413768958125897383789855088, −3.05232054536707950678398716086, −2.32306475798589800722855126549, −1.85360265533893013204759998280, 0,
1.85360265533893013204759998280, 2.32306475798589800722855126549, 3.05232054536707950678398716086, 3.76413768958125897383789855088, 4.57345750909378303031907465411, 5.28819175944476233062883916550, 6.02055038284962576508075428512, 6.84749608249199482537514238955, 7.34793214951995574771418430818
