| L(s) = 1 | + (−1.47 − 1.35i)2-s + (0.322 + 3.98i)4-s + (−9.22 + 3.35i)5-s + (−4.51 + 0.795i)7-s + (4.93 − 6.29i)8-s + (18.1 + 7.57i)10-s + (−2.67 + 7.34i)11-s + (−5.33 − 4.47i)13-s + (7.71 + 4.94i)14-s + (−15.7 + 2.57i)16-s + (7.51 − 13.0i)17-s + (18.4 − 10.6i)19-s + (−16.3 − 35.6i)20-s + (13.8 − 7.17i)22-s + (16.6 + 2.94i)23-s + ⋯ |
| L(s) = 1 | + (−0.735 − 0.677i)2-s + (0.0806 + 0.996i)4-s + (−1.84 + 0.671i)5-s + (−0.644 + 0.113i)7-s + (0.616 − 0.787i)8-s + (1.81 + 0.757i)10-s + (−0.243 + 0.668i)11-s + (−0.410 − 0.344i)13-s + (0.550 + 0.353i)14-s + (−0.986 + 0.160i)16-s + (0.441 − 0.765i)17-s + (0.970 − 0.560i)19-s + (−0.817 − 1.78i)20-s + (0.631 − 0.326i)22-s + (0.725 + 0.127i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.389586 - 0.295289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.389586 - 0.295289i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.47 + 1.35i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (9.22 - 3.35i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (4.51 - 0.795i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (2.67 - 7.34i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (5.33 + 4.47i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-7.51 + 13.0i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-18.4 + 10.6i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-16.6 - 2.94i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (14.8 - 12.4i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-14.8 - 2.61i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (14.9 - 25.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (29.3 + 24.6i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-14.2 + 39.2i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-60.2 + 10.6i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 24.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (31.1 + 85.5i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-4.84 - 27.4i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (25.2 - 30.0i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-34.2 - 19.7i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (41.3 + 71.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (10.3 + 12.3i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-55.6 - 66.3i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (-5.93 - 10.2i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-47.7 - 17.3i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−11.20688988370936082062302318906, −10.38127018300529859287696955891, −9.438665078744503815095090876306, −8.359555730149635745430575771896, −7.26798740661123882873150212267, −7.11374835583098339329276461122, −4.82462263658019984386699144990, −3.52299264130666274804745841213, −2.83497980617483699463482482282, −0.43344031571753640966934058344,
0.870519657855553041079858019307, 3.39209696935557859657279708240, 4.60129191510387848840999435143, 5.80442011246021569071203914396, 7.13551552313663794386895143466, 7.81482464892456389575253830216, 8.573036150917959365208118661675, 9.467955111208968186498131277565, 10.62189987239610044068810585191, 11.51500585698194833948226048195
