L(s) = 1 | − 0.820·2-s + 0.130·3-s − 7.32·4-s + 8.70·5-s − 0.106·6-s + 11.4·7-s + 12.5·8-s − 26.9·9-s − 7.14·10-s − 7.27·11-s − 0.952·12-s − 52.1·13-s − 9.35·14-s + 1.13·15-s + 48.3·16-s + 22.1·18-s + 84.1·19-s − 63.7·20-s + 1.48·21-s + 5.96·22-s + 157.·23-s + 1.63·24-s − 49.1·25-s + 42.7·26-s − 7.01·27-s − 83.5·28-s − 276.·29-s + ⋯ |
L(s) = 1 | − 0.290·2-s + 0.0250·3-s − 0.915·4-s + 0.778·5-s − 0.00725·6-s + 0.615·7-s + 0.555·8-s − 0.999·9-s − 0.225·10-s − 0.199·11-s − 0.0229·12-s − 1.11·13-s − 0.178·14-s + 0.0194·15-s + 0.754·16-s + 0.289·18-s + 1.01·19-s − 0.713·20-s + 0.0154·21-s + 0.0578·22-s + 1.42·23-s + 0.0139·24-s − 0.393·25-s + 0.322·26-s − 0.0500·27-s − 0.563·28-s − 1.76·29-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 | \( 1 + 0.820T + 8T^{2} \) |
| 3 | \( 1 - 0.130T + 27T^{2} \) |
| 5 | \( 1 - 8.70T + 125T^{2} \) |
| 7 | \( 1 - 11.4T + 343T^{2} \) |
| 11 | \( 1 + 7.27T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.1T + 2.19e3T^{2} \) |
| 19 | \( 1 - 84.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 276.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 235.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 352.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 90.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 167.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 589.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 288.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 519.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 557.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 304.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 316.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 66.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 118.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 215.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 931.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 402.T + 9.12e5T^{2} \) |
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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
−10.80970479432603838388414012837, −9.691485166832703605977766112965, −9.158111358158718752664643536259, −8.145122610647729335881771184148, −7.16704576104983263877089838831, −5.39702332383909838719184423683, −5.14357683494766949323042401044, −3.37916915246236202328194421329, −1.79076213772651482353341510891, 0,
1.79076213772651482353341510891, 3.37916915246236202328194421329, 5.14357683494766949323042401044, 5.39702332383909838719184423683, 7.16704576104983263877089838831, 8.145122610647729335881771184148, 9.158111358158718752664643536259, 9.691485166832703605977766112965, 10.80970479432603838388414012837