| L(s) = 1 | + 10.6i·3-s + 176. i·7-s + 129.·9-s + 400.·11-s + 784. i·13-s − 740. i·17-s − 2.77e3·19-s − 1.88e3·21-s + 176. i·23-s + 3.97e3i·27-s − 5.54e3·29-s − 5.73e3·31-s + 4.27e3i·33-s − 6.79e3i·37-s − 8.36e3·39-s + ⋯ |
| L(s) = 1 | + 0.684i·3-s + 1.36i·7-s + 0.531·9-s + 0.999·11-s + 1.28i·13-s − 0.621i·17-s − 1.76·19-s − 0.933·21-s + 0.0697i·23-s + 1.04i·27-s − 1.22·29-s − 1.07·31-s + 0.683i·33-s − 0.816i·37-s − 0.881·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.277491497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277491497\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 10.6iT - 243T^{2} \) |
| 7 | \( 1 - 176. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 400.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 784. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 740. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.77e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 176. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.79e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.40e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.33e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 279. iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.01e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.59e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.70e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.44e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 8.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.64e4iT - 8.58e9T^{2} \) |
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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
−9.638593436555775930360665839033, −9.254942306194909084489021531046, −8.685640964880491586683369438894, −7.34740642152528301817589691489, −6.44288436601998291789105566244, −5.61790171466591871642571608397, −4.44935788600233832962198564061, −3.91480495646487744649425489713, −2.45028290285046142752975695225, −1.58169306066971831523139219427,
0.26056714912685751780797558561, 1.14544025129191141674249163937, 2.10996526483419672858357471225, 3.75672667943589344932755713682, 4.18426445442351739817242905715, 5.64804446464933303338663261296, 6.64310641762200653984858274854, 7.22726007776318954551935746998, 8.006669826952895955125711787672, 8.909437001361521542596501646636
