| L(s) = 1 | + 6.46·3-s − 2.14·5-s + 20.3·7-s + 14.7·9-s − 52.0·11-s − 7.04·13-s − 13.8·15-s + 28.7·17-s − 76.4·19-s + 131.·21-s + 59.7·23-s − 120.·25-s − 79.0·27-s + 29·29-s − 3.25·31-s − 336.·33-s − 43.6·35-s − 150.·37-s − 45.5·39-s − 92.3·41-s + 100.·43-s − 31.6·45-s + 324.·47-s + 71.4·49-s + 185.·51-s − 374.·53-s + 111.·55-s + ⋯ |
| L(s) = 1 | + 1.24·3-s − 0.191·5-s + 1.09·7-s + 0.547·9-s − 1.42·11-s − 0.150·13-s − 0.238·15-s + 0.410·17-s − 0.922·19-s + 1.36·21-s + 0.541·23-s − 0.963·25-s − 0.563·27-s + 0.185·29-s − 0.0188·31-s − 1.77·33-s − 0.210·35-s − 0.669·37-s − 0.186·39-s − 0.351·41-s + 0.357·43-s − 0.104·45-s + 1.00·47-s + 0.208·49-s + 0.510·51-s − 0.971·53-s + 0.273·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 - 6.46T + 27T^{2} \) |
| 5 | \( 1 + 2.14T + 125T^{2} \) |
| 7 | \( 1 - 20.3T + 343T^{2} \) |
| 11 | \( 1 + 52.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.04T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 59.7T + 1.21e4T^{2} \) |
| 31 | \( 1 + 3.25T + 2.97e4T^{2} \) |
| 37 | \( 1 + 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 92.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 100.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 324.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 374.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 489.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 221.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 427.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 898.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.08e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 798.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 436.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 456.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 803.T + 9.12e5T^{2} \) |
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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
−8.462529586892747503017197690174, −7.73019032236109896470564432499, −7.44499635353425335410055206997, −6.00725321931339914658241704232, −5.08622840736103304014441220342, −4.31446204667673888821781915795, −3.24658036428366790454104490919, −2.44448436074356498566047688829, −1.62754872869830660272071734016, 0,
1.62754872869830660272071734016, 2.44448436074356498566047688829, 3.24658036428366790454104490919, 4.31446204667673888821781915795, 5.08622840736103304014441220342, 6.00725321931339914658241704232, 7.44499635353425335410055206997, 7.73019032236109896470564432499, 8.462529586892747503017197690174
