| L(s) = 1 | − 3.73·2-s + 3.65·3-s + 5.93·4-s + 14.5·5-s − 13.6·6-s − 12.9·7-s + 7.71·8-s − 13.6·9-s − 54.1·10-s + 20.2·11-s + 21.6·12-s − 90.7·13-s + 48.4·14-s + 53.0·15-s − 76.2·16-s + 50.9·18-s − 127.·19-s + 86.1·20-s − 47.4·21-s − 75.5·22-s + 69.5·23-s + 28.1·24-s + 85.6·25-s + 338.·26-s − 148.·27-s − 76.9·28-s + 43.9·29-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.703·3-s + 0.741·4-s + 1.29·5-s − 0.928·6-s − 0.700·7-s + 0.340·8-s − 0.505·9-s − 1.71·10-s + 0.555·11-s + 0.521·12-s − 1.93·13-s + 0.924·14-s + 0.913·15-s − 1.19·16-s + 0.666·18-s − 1.53·19-s + 0.962·20-s − 0.492·21-s − 0.732·22-s + 0.630·23-s + 0.239·24-s + 0.685·25-s + 2.55·26-s − 1.05·27-s − 0.519·28-s + 0.281·29-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 | \( 1 + 3.73T + 8T^{2} \) |
| 3 | \( 1 - 3.65T + 27T^{2} \) |
| 5 | \( 1 - 14.5T + 125T^{2} \) |
| 7 | \( 1 + 12.9T + 343T^{2} \) |
| 11 | \( 1 - 20.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 90.7T + 2.19e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 69.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 43.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 218.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 41.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 440.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 310.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 84.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 47.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 1.73T + 2.05e5T^{2} \) |
| 61 | \( 1 + 159.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 141.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 447.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 757.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 529.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 762.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 397.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 427.T + 9.12e5T^{2} \) |
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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
−10.14940503899951984848851723347, −9.941950967861671051253085896070, −9.039810975858435748446198900627, −8.416961166779896596077866575482, −7.14359246904696077256361804312, −6.26468488104413030774790881433, −4.76677495290363095934867744084, −2.79779114379239568143387187704, −1.87940627769928505063215704472, 0,
1.87940627769928505063215704472, 2.79779114379239568143387187704, 4.76677495290363095934867744084, 6.26468488104413030774790881433, 7.14359246904696077256361804312, 8.416961166779896596077866575482, 9.039810975858435748446198900627, 9.941950967861671051253085896070, 10.14940503899951984848851723347
