| L(s) = 1 | − 1.52·2-s + 3·3-s − 5.66·4-s − 19.3·5-s − 4.57·6-s − 4.84·7-s + 20.8·8-s + 9·9-s + 29.5·10-s + 61.0·11-s − 17.0·12-s + 7.39·14-s − 58.0·15-s + 13.5·16-s − 41.7·17-s − 13.7·18-s + 107.·19-s + 109.·20-s − 14.5·21-s − 93.2·22-s + 28.5·23-s + 62.5·24-s + 249.·25-s + 27·27-s + 27.4·28-s − 89.8·29-s + 88.5·30-s + ⋯ |
| L(s) = 1 | − 0.539·2-s + 0.577·3-s − 0.708·4-s − 1.72·5-s − 0.311·6-s − 0.261·7-s + 0.922·8-s + 0.333·9-s + 0.933·10-s + 1.67·11-s − 0.409·12-s + 0.141·14-s − 0.998·15-s + 0.211·16-s − 0.596·17-s − 0.179·18-s + 1.29·19-s + 1.22·20-s − 0.150·21-s − 0.903·22-s + 0.258·23-s + 0.532·24-s + 1.99·25-s + 0.192·27-s + 0.185·28-s − 0.575·29-s + 0.538·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.52T + 8T^{2} \) |
| 5 | \( 1 + 19.3T + 125T^{2} \) |
| 7 | \( 1 + 4.84T + 343T^{2} \) |
| 11 | \( 1 - 61.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 41.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 28.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 89.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 71.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 25.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 684.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 672.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 326.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 24.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 201.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 108.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−9.728765990291032714829583594327, −9.021358536086264161436550421648, −8.428771475042315446601100784615, −7.48252064690124811197209155735, −6.85711038441979435690793613026, −5.00990637960869075267089091006, −3.90245963438468504432039985416, −3.50610358684676637736631739014, −1.31101060148938842872909730105, 0,
1.31101060148938842872909730105, 3.50610358684676637736631739014, 3.90245963438468504432039985416, 5.00990637960869075267089091006, 6.85711038441979435690793613026, 7.48252064690124811197209155735, 8.428771475042315446601100784615, 9.021358536086264161436550421648, 9.728765990291032714829583594327
