| L(s) = 1 | − 404.·2-s − 1.44e4·3-s + 3.28e4·4-s + 1.27e6·5-s + 5.85e6·6-s − 5.76e6·7-s + 3.97e7·8-s + 7.98e7·9-s − 5.17e8·10-s + 9.48e8·11-s − 4.74e8·12-s − 5.26e9·13-s + 2.33e9·14-s − 1.84e10·15-s − 2.04e10·16-s + 1.14e10·17-s − 3.23e10·18-s + 4.84e10·19-s + 4.19e10·20-s + 8.33e10·21-s − 3.84e11·22-s − 2.59e11·23-s − 5.74e11·24-s + 8.68e11·25-s + 2.13e12·26-s + 7.11e11·27-s − 1.89e11·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 1.27·3-s + 0.250·4-s + 1.46·5-s + 1.42·6-s − 0.377·7-s + 0.838·8-s + 0.618·9-s − 1.63·10-s + 1.33·11-s − 0.318·12-s − 1.78·13-s + 0.422·14-s − 1.86·15-s − 1.18·16-s + 0.397·17-s − 0.691·18-s + 0.654·19-s + 0.366·20-s + 0.480·21-s − 1.49·22-s − 0.690·23-s − 1.06·24-s + 1.13·25-s + 2.00·26-s + 0.485·27-s − 0.0947·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 5.76e6T \) |
| good | 2 | \( 1 + 404.T + 1.31e5T^{2} \) |
| 3 | \( 1 + 1.44e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 1.27e6T + 7.62e11T^{2} \) |
| 11 | \( 1 - 9.48e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 5.26e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.14e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 4.84e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 2.59e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 5.77e11T + 7.25e24T^{2} \) |
| 31 | \( 1 + 3.05e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.74e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 6.58e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 5.34e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.05e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 5.93e13T + 2.05e29T^{2} \) |
| 59 | \( 1 - 8.69e13T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.53e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 9.46e14T + 1.10e31T^{2} \) |
| 71 | \( 1 - 4.24e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 3.23e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.73e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.79e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.32e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 9.03e16T + 5.95e33T^{2} \) |
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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
−17.14081394877282172458351003548, −16.82999924008839660226768320723, −14.03979660254431935265374856065, −12.09112476852973258962726564494, −10.23497381352073843148526902452, −9.389885379481557048368265100928, −6.81346863017482316365978394549, −5.25566293256639367013452505876, −1.59257517371165018500067054973, 0,
1.59257517371165018500067054973, 5.25566293256639367013452505876, 6.81346863017482316365978394549, 9.389885379481557048368265100928, 10.23497381352073843148526902452, 12.09112476852973258962726564494, 14.03979660254431935265374856065, 16.82999924008839660226768320723, 17.14081394877282172458351003548
