| L(s) = 1 | + 40.9·2-s + 105.·3-s + 1.16e3·4-s − 2.07e3·5-s + 4.30e3·6-s − 5.18e3·7-s + 2.67e4·8-s − 8.65e3·9-s − 8.48e4·10-s + 8.39e4·11-s + 1.22e5·12-s − 3.84e4·13-s − 2.12e5·14-s − 2.17e5·15-s + 4.99e5·16-s − 3.54e5·18-s + 7.27e5·19-s − 2.41e6·20-s − 5.44e5·21-s + 3.43e6·22-s − 2.28e6·23-s + 2.81e6·24-s + 2.34e6·25-s − 1.57e6·26-s − 2.97e6·27-s − 6.03e6·28-s − 2.05e6·29-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 0.748·3-s + 2.27·4-s − 1.48·5-s + 1.35·6-s − 0.815·7-s + 2.31·8-s − 0.439·9-s − 2.68·10-s + 1.72·11-s + 1.70·12-s − 0.373·13-s − 1.47·14-s − 1.11·15-s + 1.90·16-s − 0.795·18-s + 1.28·19-s − 3.37·20-s − 0.610·21-s + 3.12·22-s − 1.70·23-s + 1.72·24-s + 1.19·25-s − 0.675·26-s − 1.07·27-s − 1.85·28-s − 0.540·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 40.9T + 512T^{2} \) |
| 3 | \( 1 - 105.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.07e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.18e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.39e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.84e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 7.27e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.28e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.05e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.10e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.39e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.53e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.22e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.95e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.74e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.17e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.17e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.00e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.31e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.21e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.05e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.83e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.76e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.15e8T + 7.60e17T^{2} \) |
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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
−9.813612649953362459273704218506, −8.610885234371098085654400699355, −7.48159013992972679088242626921, −6.74099967321163431373614032644, −5.67150414020450474590471503089, −4.35111523383035625125982412821, −3.46745185932414315268998645936, −3.33508216738957201049665368633, −1.80941908570961369265864890918, 0,
1.80941908570961369265864890918, 3.33508216738957201049665368633, 3.46745185932414315268998645936, 4.35111523383035625125982412821, 5.67150414020450474590471503089, 6.74099967321163431373614032644, 7.48159013992972679088242626921, 8.610885234371098085654400699355, 9.813612649953362459273704218506
