| L(s) = 1 | − 0.610·2-s − 1.93·3-s − 1.62·4-s + 1.75·5-s + 1.17·6-s + 7-s + 2.21·8-s + 0.726·9-s − 1.07·10-s + 11-s + 3.14·12-s + 13-s − 0.610·14-s − 3.38·15-s + 1.90·16-s − 6.96·17-s − 0.443·18-s + 3.17·19-s − 2.85·20-s − 1.93·21-s − 0.610·22-s − 7.45·23-s − 4.27·24-s − 1.91·25-s − 0.610·26-s + 4.38·27-s − 1.62·28-s + ⋯ |
| L(s) = 1 | − 0.431·2-s − 1.11·3-s − 0.813·4-s + 0.785·5-s + 0.481·6-s + 0.377·7-s + 0.783·8-s + 0.242·9-s − 0.339·10-s + 0.301·11-s + 0.906·12-s + 0.277·13-s − 0.163·14-s − 0.875·15-s + 0.475·16-s − 1.68·17-s − 0.104·18-s + 0.728·19-s − 0.638·20-s − 0.421·21-s − 0.130·22-s − 1.55·23-s − 0.872·24-s − 0.383·25-s − 0.119·26-s + 0.844·27-s − 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7366968175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7366968175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.610T + 2T^{2} \) |
| 3 | \( 1 + 1.93T + 3T^{2} \) |
| 5 | \( 1 - 1.75T + 5T^{2} \) |
| 17 | \( 1 + 6.96T + 17T^{2} \) |
| 19 | \( 1 - 3.17T + 19T^{2} \) |
| 23 | \( 1 + 7.45T + 23T^{2} \) |
| 29 | \( 1 - 8.40T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 - 8.28T + 43T^{2} \) |
| 47 | \( 1 - 7.74T + 47T^{2} \) |
| 53 | \( 1 - 9.67T + 53T^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 + 4.81T + 61T^{2} \) |
| 67 | \( 1 + 3.73T + 67T^{2} \) |
| 71 | \( 1 + 0.781T + 71T^{2} \) |
| 73 | \( 1 + 0.513T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 4.71T + 83T^{2} \) |
| 89 | \( 1 + 4.72T + 89T^{2} \) |
| 97 | \( 1 - 9.98T + 97T^{2} \) |
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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
−10.11071104356279554291461035325, −9.118416593444809878117306502667, −8.565276122533266884700852998156, −7.47341247403809613103868001167, −6.33608806352328576706048209824, −5.73817326862573462859482397799, −4.83044436696476323650437652575, −4.04150317802333784167392423576, −2.16900497225694208334074876517, −0.76566963362268650323458884910,
0.76566963362268650323458884910, 2.16900497225694208334074876517, 4.04150317802333784167392423576, 4.83044436696476323650437652575, 5.73817326862573462859482397799, 6.33608806352328576706048209824, 7.47341247403809613103868001167, 8.565276122533266884700852998156, 9.118416593444809878117306502667, 10.11071104356279554291461035325
