| L(s) = 1 | − 0.463i·2-s + (−38.1 + 71.4i)3-s + 255.·4-s − 649. i·5-s + (33.1 + 17.6i)6-s − 831.·7-s − 237. i·8-s + (−3.65e3 − 5.45e3i)9-s − 301.·10-s − 4.41e3i·11-s + (−9.75e3 + 1.82e4i)12-s + 4.31e4·13-s + 385. i·14-s + (4.64e4 + 2.47e4i)15-s + 6.53e4·16-s − 1.04e5i·17-s + ⋯ |
| L(s) = 1 | − 0.0289i·2-s + (−0.470 + 0.882i)3-s + 0.999·4-s − 1.03i·5-s + (0.0255 + 0.0136i)6-s − 0.346·7-s − 0.0579i·8-s + (−0.556 − 0.830i)9-s − 0.0301·10-s − 0.301i·11-s + (−0.470 + 0.881i)12-s + 1.51·13-s + 0.0100i·14-s + (0.917 + 0.489i)15-s + 0.997·16-s − 1.25i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.82835 - 0.457376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82835 - 0.457376i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (38.1 - 71.4i)T \) |
| 11 | \( 1 + 4.41e3iT \) |
| good | 2 | \( 1 + 0.463iT - 256T^{2} \) |
| 5 | \( 1 + 649. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 831.T + 5.76e6T^{2} \) |
| 13 | \( 1 - 4.31e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.04e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 7.59e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 3.18e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 3.35e4iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.04e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 4.75e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 9.58e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.21e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 4.32e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.46e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.53e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 6.61e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 8.92e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.10e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 8.31e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 7.01e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 7.67e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 8.67e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.38e8T + 7.83e15T^{2} \) |
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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
−15.31652914210545665864967361302, −13.54619952511556304514513424262, −12.02580554135564932630353279301, −11.19092047794588733577701763188, −9.830407545734046210057734377064, −8.501137932022704566646202582298, −6.47010913717835449230047257939, −5.15870635109652044622778798961, −3.35914017638414182799094663904, −0.923287696548097197807068040388,
1.52164996925525145505290291383, 3.10818861887179783689040034901, 6.07498688671137360043221717471, 6.74075323253246330560390317376, 8.063540863449982906904960891064, 10.47520434770432758018738958561, 11.23267822592425947963088979494, 12.42868230898041159043771476840, 13.73614887713319965770302124006, 15.09136693131975964457645553884
