| L(s) = 1 | − 2·2-s − 3.66·3-s + 4·4-s − 8.53·5-s + 7.32·6-s + 4.20·7-s − 8·8-s − 13.5·9-s + 17.0·10-s + 65.3·11-s − 14.6·12-s − 8.40·14-s + 31.2·15-s + 16·16-s − 26.9·17-s + 27.1·18-s + 13.3·19-s − 34.1·20-s − 15.3·21-s − 130.·22-s + 159.·23-s + 29.3·24-s − 52.2·25-s + 148.·27-s + 16.8·28-s − 301.·29-s − 62.5·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.705·3-s + 0.5·4-s − 0.763·5-s + 0.498·6-s + 0.226·7-s − 0.353·8-s − 0.502·9-s + 0.539·10-s + 1.79·11-s − 0.352·12-s − 0.160·14-s + 0.538·15-s + 0.250·16-s − 0.383·17-s + 0.355·18-s + 0.161·19-s − 0.381·20-s − 0.159·21-s − 1.26·22-s + 1.44·23-s + 0.249·24-s − 0.417·25-s + 1.05·27-s + 0.113·28-s − 1.92·29-s − 0.380·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3.66T + 27T^{2} \) |
| 5 | \( 1 + 8.53T + 125T^{2} \) |
| 7 | \( 1 - 4.20T + 343T^{2} \) |
| 11 | \( 1 - 65.3T + 1.33e3T^{2} \) |
| 17 | \( 1 + 26.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 159.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 301.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 73.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 118.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 432.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 356.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 588.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 269.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 230.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 380.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 435.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 65.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 885.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 385.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 254.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 372.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.31e3T + 9.12e5T^{2} \) |
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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
−11.04223386526110528394225604320, −9.548517340320306626083237424107, −8.886038245653125739258569205241, −7.82422861061348441504454343001, −6.81826162437603406959095274502, −5.96832179607908134765538406426, −4.58520609748213558929223606343, −3.31364193737391448873116149553, −1.40132074954442257964202334784, 0,
1.40132074954442257964202334784, 3.31364193737391448873116149553, 4.58520609748213558929223606343, 5.96832179607908134765538406426, 6.81826162437603406959095274502, 7.82422861061348441504454343001, 8.886038245653125739258569205241, 9.548517340320306626083237424107, 11.04223386526110528394225604320
