| L(s) = 1 | + (0.307 + 0.177i)2-s + (8.32 + 3.42i)3-s + (−7.93 − 13.7i)4-s + (−30.0 + 17.3i)5-s + (1.95 + 2.53i)6-s + (15.6 − 27.0i)7-s − 11.3i·8-s + (57.5 + 57.0i)9-s − 12.3·10-s + (49.9 + 28.8i)11-s + (−18.9 − 141. i)12-s + (36.6 + 63.4i)13-s + (9.60 − 5.54i)14-s + (−309. + 41.4i)15-s + (−124. + 216. i)16-s − 386. i·17-s + ⋯ |
| L(s) = 1 | + (0.0769 + 0.0443i)2-s + (0.924 + 0.380i)3-s + (−0.496 − 0.859i)4-s + (−1.20 + 0.694i)5-s + (0.0541 + 0.0703i)6-s + (0.318 − 0.551i)7-s − 0.176i·8-s + (0.709 + 0.704i)9-s − 0.123·10-s + (0.413 + 0.238i)11-s + (−0.131 − 0.983i)12-s + (0.216 + 0.375i)13-s + (0.0489 − 0.0282i)14-s + (−1.37 + 0.184i)15-s + (−0.488 + 0.845i)16-s − 1.33i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0910i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 - 0.0910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.08483 + 0.0495168i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08483 + 0.0495168i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.32 - 3.42i)T \) |
| good | 2 | \( 1 + (-0.307 - 0.177i)T + (8 + 13.8i)T^{2} \) |
| 5 | \( 1 + (30.0 - 17.3i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (-15.6 + 27.0i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-49.9 - 28.8i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-36.6 - 63.4i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + 386. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 115.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (474. - 274. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (680. + 392. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (272. + 471. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 898.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (-2.24e3 + 1.29e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.00e3 - 1.73e3i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (702. + 405. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 - 2.22e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-1.30e3 + 756. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-951. + 1.64e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (2.25e3 + 3.90e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 3.99e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.43e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (601. - 1.04e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-8.01e3 - 4.62e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + 8.92e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (3.33e3 - 5.77e3i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
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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
−20.38218719415397216516816675543, −19.44949698332981315930654600903, −18.38973112605092850225766113355, −15.91558211217783749296839207968, −14.76810288946263150734570503419, −13.81052998359534188017306493281, −11.20679484717477736322310650472, −9.535298948836360342899739804578, −7.53365314695087897131605859684, −4.14130148445312395828322151129,
3.88336784903434659782190843136, 7.922935610310642112929559617236, 8.767143711347760277045197583229, 11.98078364337353344745302622226, 12.98503970403366044379111634287, 14.76834799093924412021759712038, 16.27823872271238517322629403978, 18.05861278404283201477158392278, 19.46136571417082550534379139040, 20.57085023065984543640927923438
