| L(s) = 1 | + 2-s − 0.551·3-s + 4-s + 1.55·5-s − 0.551·6-s − 7-s + 8-s − 2.69·9-s + 1.55·10-s + 3.69·11-s − 0.551·12-s − 4.69·13-s − 14-s − 0.856·15-s + 16-s + 2.24·17-s − 2.69·18-s + 3.14·19-s + 1.55·20-s + 0.551·21-s + 3.69·22-s − 0.551·24-s − 2.59·25-s − 4.69·26-s + 3.14·27-s − 28-s − 8.14·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.318·3-s + 0.5·4-s + 0.694·5-s − 0.225·6-s − 0.377·7-s + 0.353·8-s − 0.898·9-s + 0.490·10-s + 1.11·11-s − 0.159·12-s − 1.30·13-s − 0.267·14-s − 0.221·15-s + 0.250·16-s + 0.545·17-s − 0.635·18-s + 0.721·19-s + 0.347·20-s + 0.120·21-s + 0.787·22-s − 0.112·24-s − 0.518·25-s − 0.920·26-s + 0.604·27-s − 0.188·28-s − 1.51·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.551T + 3T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 13 | \( 1 + 4.69T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 29 | \( 1 + 8.14T + 29T^{2} \) |
| 31 | \( 1 + 3.14T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 + 7.14T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 8.35T + 53T^{2} \) |
| 59 | \( 1 - 7.39T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 2.20T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 - 7.63T + 79T^{2} \) |
| 83 | \( 1 + 0.799T + 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−7.32882596126161642908395941908, −6.69965531637372083979652321673, −6.03927835975592187932666947054, −5.38472438513691278215712649868, −5.01009585774606498603141833360, −3.81846518853179607319673319599, −3.27377004508679129767881557384, −2.32158365766062229403439902028, −1.50838745345643818061093757995, 0,
1.50838745345643818061093757995, 2.32158365766062229403439902028, 3.27377004508679129767881557384, 3.81846518853179607319673319599, 5.01009585774606498603141833360, 5.38472438513691278215712649868, 6.03927835975592187932666947054, 6.69965531637372083979652321673, 7.32882596126161642908395941908
