| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s − 19.5·5-s − 8·6-s − 7·7-s + 8·8-s − 11·9-s − 39.1·10-s − 23.1·11-s − 16·12-s − 14·14-s + 78.3·15-s + 16·16-s − 2.80·17-s − 22·18-s + 99.1·19-s − 78.3·20-s + 28·21-s − 46.3·22-s + 69.5·23-s − 32·24-s + 258.·25-s + 152·27-s − 28·28-s + 187.·29-s + 156.·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 0.5·4-s − 1.75·5-s − 0.544·6-s − 0.377·7-s + 0.353·8-s − 0.407·9-s − 1.23·10-s − 0.635·11-s − 0.384·12-s − 0.267·14-s + 1.34·15-s + 0.250·16-s − 0.0400·17-s − 0.288·18-s + 1.19·19-s − 0.876·20-s + 0.290·21-s − 0.449·22-s + 0.630·23-s − 0.272·24-s + 2.07·25-s + 1.08·27-s − 0.188·28-s + 1.20·29-s + 0.954·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 4T + 27T^{2} \) |
| 5 | \( 1 + 19.5T + 125T^{2} \) |
| 11 | \( 1 + 23.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 2.80T + 4.91e3T^{2} \) |
| 19 | \( 1 - 99.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 69.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 267.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 389.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 355.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 720.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 601.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 224.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 257.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 582.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 424.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 501.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.60e3T + 9.12e5T^{2} \) |
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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
−8.186270974943796645155878109223, −7.08762417797894855740810898998, −6.95220269372537596223137354935, −5.57729618711272406085822116873, −5.17508233011128265670205156571, −4.22149319887176056958981587136, −3.40869575844716598984813357735, −2.74046101809348913793398766948, −0.912627117124973737855348940271, 0,
0.912627117124973737855348940271, 2.74046101809348913793398766948, 3.40869575844716598984813357735, 4.22149319887176056958981587136, 5.17508233011128265670205156571, 5.57729618711272406085822116873, 6.95220269372537596223137354935, 7.08762417797894855740810898998, 8.186270974943796645155878109223
