| L(s) = 1 | − 2·2-s + 0.405·3-s + 4·4-s + 6.36·5-s − 0.811·6-s − 2.55·7-s − 8·8-s − 26.8·9-s − 12.7·10-s + 26.1·11-s + 1.62·12-s + 5.10·14-s + 2.58·15-s + 16·16-s − 93.7·17-s + 53.6·18-s − 37.2·19-s + 25.4·20-s − 1.03·21-s − 52.2·22-s − 104.·23-s − 3.24·24-s − 84.5·25-s − 21.8·27-s − 10.2·28-s + 249.·29-s − 5.16·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0780·3-s + 0.5·4-s + 0.569·5-s − 0.0552·6-s − 0.137·7-s − 0.353·8-s − 0.993·9-s − 0.402·10-s + 0.715·11-s + 0.0390·12-s + 0.0973·14-s + 0.0444·15-s + 0.250·16-s − 1.33·17-s + 0.702·18-s − 0.449·19-s + 0.284·20-s − 0.0107·21-s − 0.506·22-s − 0.951·23-s − 0.0276·24-s − 0.676·25-s − 0.155·27-s − 0.0688·28-s + 1.59·29-s − 0.0314·30-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 | \( 1 - 0.405T + 27T^{2} \) |
| 5 | \( 1 - 6.36T + 125T^{2} \) |
| 7 | \( 1 + 2.55T + 343T^{2} \) |
| 11 | \( 1 - 26.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 93.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 249.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 278.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 10.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 238.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 424.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 774.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 123.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 881.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 118.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 209.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 532.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 376.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 42.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 639.T + 9.12e5T^{2} \) |
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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
−10.48507400452114034455344179041, −9.688010731060488302240715097642, −8.742054684412510920450851833998, −8.139381257562754021363752272394, −6.58389709471778843864988520433, −6.14345367119653406161522408975, −4.59548535936550969534478016969, −2.99076026862635851518010660339, −1.78465533015960456163356550174, 0,
1.78465533015960456163356550174, 2.99076026862635851518010660339, 4.59548535936550969534478016969, 6.14345367119653406161522408975, 6.58389709471778843864988520433, 8.139381257562754021363752272394, 8.742054684412510920450851833998, 9.688010731060488302240715097642, 10.48507400452114034455344179041
