| L(s) = 1 | + 650.·2-s + 5.12e3·3-s + 2.91e5·4-s + 4.70e4·5-s + 3.33e6·6-s + 1.49e7·7-s + 1.04e8·8-s − 1.02e8·9-s + 3.05e7·10-s + 2.14e8·11-s + 1.49e9·12-s + 1.94e9·13-s + 9.73e9·14-s + 2.40e8·15-s + 2.96e10·16-s + 1.22e10·17-s − 6.68e10·18-s − 9.27e10·19-s + 1.37e10·20-s + 7.67e10·21-s + 1.39e11·22-s + 1.36e11·23-s + 5.35e11·24-s − 7.60e11·25-s + 1.26e12·26-s − 1.18e12·27-s + 4.36e12·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 0.451·3-s + 2.22·4-s + 0.0538·5-s + 0.810·6-s + 0.981·7-s + 2.20·8-s − 0.796·9-s + 0.0966·10-s + 0.301·11-s + 1.00·12-s + 0.661·13-s + 1.76·14-s + 0.0242·15-s + 1.72·16-s + 0.426·17-s − 1.43·18-s − 1.25·19-s + 0.119·20-s + 0.442·21-s + 0.541·22-s + 0.363·23-s + 0.992·24-s − 0.997·25-s + 1.18·26-s − 0.810·27-s + 2.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(6.672245923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.672245923\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 2.14e8T \) |
| good | 2 | \( 1 - 650.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 5.12e3T + 1.29e8T^{2} \) |
| 5 | \( 1 - 4.70e4T + 7.62e11T^{2} \) |
| 7 | \( 1 - 1.49e7T + 2.32e14T^{2} \) |
| 13 | \( 1 - 1.94e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.22e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 9.27e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.36e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 3.99e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 2.19e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.76e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 4.67e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 4.77e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.57e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 6.16e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 2.38e13T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.28e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 5.99e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 3.74e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.18e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 6.21e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.88e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 2.86e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 7.24e16T + 5.95e33T^{2} \) |
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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
−15.53859744714918531967833756078, −14.44151050907862987971213130091, −13.70886659811377025278551881592, −12.12965534002289591430711898187, −10.98088180252064421902467818032, −8.272652136392744128113406045016, −6.31119011424990291746514553310, −4.86906130566053797702250063585, −3.41881222427643902711636840728, −1.91887945815520710234194956400,
1.91887945815520710234194956400, 3.41881222427643902711636840728, 4.86906130566053797702250063585, 6.31119011424990291746514553310, 8.272652136392744128113406045016, 10.98088180252064421902467818032, 12.12965534002289591430711898187, 13.70886659811377025278551881592, 14.44151050907862987971213130091, 15.53859744714918531967833756078
