| L(s) = 1 | − 1.70·2-s − 4.44·3-s − 5.09·4-s + 6.80·5-s + 7.57·6-s − 13.8·7-s + 22.3·8-s − 7.25·9-s − 11.6·10-s + 30.8·11-s + 22.6·12-s + 66.0·13-s + 23.6·14-s − 30.2·15-s + 2.65·16-s + 12.3·18-s − 79.7·19-s − 34.6·20-s + 61.6·21-s − 52.5·22-s + 28.8·23-s − 99.2·24-s − 78.6·25-s − 112.·26-s + 152.·27-s + 70.6·28-s + 266.·29-s + ⋯ |
| L(s) = 1 | − 0.602·2-s − 0.855·3-s − 0.636·4-s + 0.609·5-s + 0.515·6-s − 0.749·7-s + 0.986·8-s − 0.268·9-s − 0.367·10-s + 0.845·11-s + 0.544·12-s + 1.40·13-s + 0.451·14-s − 0.520·15-s + 0.0414·16-s + 0.161·18-s − 0.962·19-s − 0.387·20-s + 0.640·21-s − 0.509·22-s + 0.261·23-s − 0.843·24-s − 0.629·25-s − 0.849·26-s + 1.08·27-s + 0.476·28-s + 1.70·29-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 | \( 1 + 1.70T + 8T^{2} \) |
| 3 | \( 1 + 4.44T + 27T^{2} \) |
| 5 | \( 1 - 6.80T + 125T^{2} \) |
| 7 | \( 1 + 13.8T + 343T^{2} \) |
| 11 | \( 1 - 30.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.0T + 2.19e3T^{2} \) |
| 19 | \( 1 + 79.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 28.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 266.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 35.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 32.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 516.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 210.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 87.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 310.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 365.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 660.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 398.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 643.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 384.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 153.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 599.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 44.5T + 9.12e5T^{2} \) |
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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
−10.67880451613082199049162961260, −10.02315558165505941520181601223, −8.963335919551574303765634476015, −8.388323724117714228380081402940, −6.63890457773248826876945315245, −6.08588334053424720351657457051, −4.85972602121662609505771472768, −3.52843057022553843148639464275, −1.39811742401744527132762499171, 0,
1.39811742401744527132762499171, 3.52843057022553843148639464275, 4.85972602121662609505771472768, 6.08588334053424720351657457051, 6.63890457773248826876945315245, 8.388323724117714228380081402940, 8.963335919551574303765634476015, 10.02315558165505941520181601223, 10.67880451613082199049162961260
