| L(s) = 1 | − 3.49·2-s − 2.82·3-s + 4.24·4-s − 8.71·5-s + 9.88·6-s − 6.85·7-s + 13.1·8-s − 19.0·9-s + 30.5·10-s + 61.7·11-s − 12.0·12-s − 5.37·13-s + 23.9·14-s + 24.6·15-s − 79.9·16-s + 66.5·18-s + 96.7·19-s − 37.0·20-s + 19.3·21-s − 216.·22-s + 116.·23-s − 37.1·24-s − 48.9·25-s + 18.8·26-s + 130.·27-s − 29.1·28-s − 197.·29-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 0.543·3-s + 0.530·4-s − 0.779·5-s + 0.672·6-s − 0.370·7-s + 0.580·8-s − 0.704·9-s + 0.964·10-s + 1.69·11-s − 0.288·12-s − 0.114·13-s + 0.457·14-s + 0.424·15-s − 1.24·16-s + 0.871·18-s + 1.16·19-s − 0.414·20-s + 0.201·21-s − 2.09·22-s + 1.05·23-s − 0.315·24-s − 0.391·25-s + 0.141·26-s + 0.926·27-s − 0.196·28-s − 1.26·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 + 3.49T + 8T^{2} \) |
| 3 | \( 1 + 2.82T + 27T^{2} \) |
| 5 | \( 1 + 8.71T + 125T^{2} \) |
| 7 | \( 1 + 6.85T + 343T^{2} \) |
| 11 | \( 1 - 61.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.37T + 2.19e3T^{2} \) |
| 19 | \( 1 - 96.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 197.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 138.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 111.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 166.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 165.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 714.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 846.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 4.99T + 2.26e5T^{2} \) |
| 67 | \( 1 + 314.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 118.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 650.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 208.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 742.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 215.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 705.T + 9.12e5T^{2} \) |
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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
−11.05833038688375121781237685534, −9.644244063259159494134241134526, −9.161756039500948713913046236075, −8.109054026243371235941117090762, −7.18305340785982463495027363709, −6.19040218441307995350188259625, −4.70513753394617677829617126161, −3.37053713058367719653000290860, −1.24481141146538019570351685906, 0,
1.24481141146538019570351685906, 3.37053713058367719653000290860, 4.70513753394617677829617126161, 6.19040218441307995350188259625, 7.18305340785982463495027363709, 8.109054026243371235941117090762, 9.161756039500948713913046236075, 9.644244063259159494134241134526, 11.05833038688375121781237685534
