| L(s) = 1 | + 16·2-s + 240.·3-s + 256·4-s − 622.·5-s + 3.85e3·6-s − 1.01e4·7-s + 4.09e3·8-s + 3.83e4·9-s − 9.96e3·10-s − 3.09e3·11-s + 6.16e4·12-s − 1.61e5·14-s − 1.50e5·15-s + 6.55e4·16-s − 2.63e5·17-s + 6.13e5·18-s + 5.27e5·19-s − 1.59e5·20-s − 2.43e6·21-s − 4.95e4·22-s + 1.89e6·23-s + 9.86e5·24-s − 1.56e6·25-s + 4.50e6·27-s − 2.58e6·28-s − 4.41e6·29-s − 2.40e6·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.71·3-s + 0.5·4-s − 0.445·5-s + 1.21·6-s − 1.59·7-s + 0.353·8-s + 1.94·9-s − 0.315·10-s − 0.0637·11-s + 0.858·12-s − 1.12·14-s − 0.765·15-s + 0.250·16-s − 0.764·17-s + 1.37·18-s + 0.928·19-s − 0.222·20-s − 2.73·21-s − 0.0450·22-s + 1.41·23-s + 0.607·24-s − 0.801·25-s + 1.63·27-s − 0.795·28-s − 1.15·29-s − 0.541·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 240.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 622.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.01e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.09e3T + 2.35e9T^{2} \) |
| 17 | \( 1 + 2.63e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.27e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.89e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.41e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.81e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.28e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.08e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.93e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.78e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.23e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.66e4T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.80e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.86e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.61e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.11e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.10e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.65e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.97e8T + 7.60e17T^{2} \) |
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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
−9.334590923871474525480243695151, −8.771495804740280799487505006976, −7.37558617096884798737812638256, −7.05028536947065680223560958089, −5.61750915012586520836880218465, −4.11947118373341757537485467031, −3.40174600400432903921925728661, −2.85226297050499863679895496632, −1.69823388249055958804210361290, 0,
1.69823388249055958804210361290, 2.85226297050499863679895496632, 3.40174600400432903921925728661, 4.11947118373341757537485467031, 5.61750915012586520836880218465, 7.05028536947065680223560958089, 7.37558617096884798737812638256, 8.771495804740280799487505006976, 9.334590923871474525480243695151
