| L(s) = 1 | + (−1.71 + 1.03i)2-s + (1.87 − 3.53i)4-s + 0.169i·5-s − 4.44i·7-s + (0.423 + 7.98i)8-s + (−0.174 − 0.289i)10-s − 7.38·11-s − 11.7i·13-s + (4.57 + 7.61i)14-s + (−8.95 − 13.2i)16-s − 10.7·17-s − 16.1·19-s + (0.597 + 0.317i)20-s + (12.6 − 7.61i)22-s − 21.0i·23-s + ⋯ |
| L(s) = 1 | + (−0.857 + 0.515i)2-s + (0.469 − 0.883i)4-s + 0.0338i·5-s − 0.634i·7-s + (0.0529 + 0.998i)8-s + (−0.0174 − 0.0289i)10-s − 0.671·11-s − 0.901i·13-s + (0.326 + 0.543i)14-s + (−0.559 − 0.828i)16-s − 0.631·17-s − 0.850·19-s + (0.0298 + 0.0158i)20-s + (0.575 − 0.345i)22-s − 0.916i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0529 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0529 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.467256 - 0.443147i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.467256 - 0.443147i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.71 - 1.03i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.169iT - 25T^{2} \) |
| 7 | \( 1 + 4.44iT - 49T^{2} \) |
| 11 | \( 1 + 7.38T + 121T^{2} \) |
| 13 | \( 1 + 11.7iT - 169T^{2} \) |
| 17 | \( 1 + 10.7T + 289T^{2} \) |
| 19 | \( 1 + 16.1T + 361T^{2} \) |
| 23 | \( 1 + 21.0iT - 529T^{2} \) |
| 29 | \( 1 + 48.9iT - 841T^{2} \) |
| 31 | \( 1 + 42.8iT - 961T^{2} \) |
| 37 | \( 1 - 31.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 8.47T + 1.68e3T^{2} \) |
| 43 | \( 1 + 19.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 71.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 9.55iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 48.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 101. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 120.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 84.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 92.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 70.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 2.94T + 6.88e3T^{2} \) |
| 89 | \( 1 + 98.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 115.T + 9.40e3T^{2} \) |
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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
−11.54627361739861316768520606894, −10.53670405184109969480774716177, −10.06213073340046914916222375693, −8.688319890916132095962119031686, −7.937976230854575196782309547987, −6.90326691692930947278580689101, −5.85931562280210856119316859172, −4.51018972025911209883493247686, −2.47774917486522381446261117397, −0.45462071000531988272053467887,
1.81449574126953215313494602952, 3.17227362588093501833434757062, 4.79992513205270892189677413192, 6.43144055725208027723422040984, 7.45225305614179405003712383117, 8.733884831928763416812757821359, 9.165025002385464752598406549936, 10.52187009620833309649037550199, 11.13055886423804087228896060489, 12.27800404024317172364761037526
