| L(s) = 1 | − 114.·2-s − 157.·3-s + 5.02e3·4-s + 5.79e4·5-s + 1.80e4·6-s + 4.75e5·7-s + 3.64e5·8-s − 1.56e6·9-s − 6.66e6·10-s + 1.01e7·11-s − 7.89e5·12-s − 5.77e6·13-s − 5.47e7·14-s − 9.11e6·15-s − 8.30e7·16-s − 4.86e6·17-s + 1.80e8·18-s − 5.91e7·19-s + 2.91e8·20-s − 7.48e7·21-s − 1.16e9·22-s + 1.48e8·23-s − 5.72e7·24-s + 2.13e9·25-s + 6.64e8·26-s + 4.97e8·27-s + 2.39e9·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.124·3-s + 0.613·4-s + 1.65·5-s + 0.158·6-s + 1.52·7-s + 0.491·8-s − 0.984·9-s − 2.10·10-s + 1.72·11-s − 0.0763·12-s − 0.332·13-s − 1.94·14-s − 0.206·15-s − 1.23·16-s − 0.0489·17-s + 1.25·18-s − 0.288·19-s + 1.01·20-s − 0.190·21-s − 2.19·22-s + 0.208·23-s − 0.0611·24-s + 1.75·25-s + 0.421·26-s + 0.247·27-s + 0.937·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.564307032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564307032\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 1.48e8T \) |
| good | 2 | \( 1 + 114.T + 8.19e3T^{2} \) |
| 3 | \( 1 + 157.T + 1.59e6T^{2} \) |
| 5 | \( 1 - 5.79e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 4.75e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 1.01e7T + 3.45e13T^{2} \) |
| 13 | \( 1 + 5.77e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 4.86e6T + 9.90e15T^{2} \) |
| 19 | \( 1 + 5.91e7T + 4.20e16T^{2} \) |
| 29 | \( 1 - 3.97e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 1.85e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.31e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 1.47e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 3.13e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 9.37e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.27e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 4.38e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 6.31e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.22e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 4.86e9T + 1.16e24T^{2} \) |
| 73 | \( 1 + 9.83e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.61e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.33e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.12e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 9.78e12T + 6.73e25T^{2} \) |
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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
−14.46872538009324103152452425274, −13.91243085932978452829756207614, −11.70915154236572954772774797161, −10.51656651011143387658866742439, −9.250293621062105856232809017134, −8.424798081512658609740764892366, −6.54852894477787332269275227817, −4.96916832727626829673721908510, −2.02494326091313054272816606389, −1.12048497487294035337210581563,
1.12048497487294035337210581563, 2.02494326091313054272816606389, 4.96916832727626829673721908510, 6.54852894477787332269275227817, 8.424798081512658609740764892366, 9.250293621062105856232809017134, 10.51656651011143387658866742439, 11.70915154236572954772774797161, 13.91243085932978452829756207614, 14.46872538009324103152452425274
