| L(s) = 1 | + 2.30·2-s + 4.54·3-s − 2.69·4-s + 4.16·5-s + 10.4·6-s + 17.3·7-s − 24.6·8-s − 6.31·9-s + 9.59·10-s − 11·11-s − 12.2·12-s + 39.9·14-s + 18.9·15-s − 35.1·16-s + 1.39·17-s − 14.5·18-s − 58.9·19-s − 11.2·20-s + 78.9·21-s − 25.3·22-s + 46.6·23-s − 112.·24-s − 107.·25-s − 151.·27-s − 46.8·28-s + 175.·29-s + 43.6·30-s + ⋯ |
| L(s) = 1 | + 0.814·2-s + 0.875·3-s − 0.337·4-s + 0.372·5-s + 0.712·6-s + 0.937·7-s − 1.08·8-s − 0.233·9-s + 0.303·10-s − 0.301·11-s − 0.295·12-s + 0.763·14-s + 0.326·15-s − 0.548·16-s + 0.0198·17-s − 0.190·18-s − 0.711·19-s − 0.125·20-s + 0.820·21-s − 0.245·22-s + 0.422·23-s − 0.952·24-s − 0.861·25-s − 1.07·27-s − 0.316·28-s + 1.12·29-s + 0.265·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.30T + 8T^{2} \) |
| 3 | \( 1 - 4.54T + 27T^{2} \) |
| 5 | \( 1 - 4.16T + 125T^{2} \) |
| 7 | \( 1 - 17.3T + 343T^{2} \) |
| 17 | \( 1 - 1.39T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 46.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 33.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 376.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 270.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 562.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 356.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 138.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 358.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 291.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 90.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 366.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 270.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 746.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 869.T + 9.12e5T^{2} \) |
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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
−8.328233533810927181728073915769, −8.083569678810670858936521014432, −6.78559992190276303349938837998, −5.85457073418438657485443515007, −5.09279537417072453553555738147, −4.39738544846530001953675589684, −3.42479103593178067151506518103, −2.63122548914633838450505543583, −1.64216054188870969321051858664, 0,
1.64216054188870969321051858664, 2.63122548914633838450505543583, 3.42479103593178067151506518103, 4.39738544846530001953675589684, 5.09279537417072453553555738147, 5.85457073418438657485443515007, 6.78559992190276303349938837998, 8.083569678810670858936521014432, 8.328233533810927181728073915769
