| L(s) = 1 | + (0.132 + 0.408i)2-s + (6.32 − 4.59i)4-s + (−11.1 − 0.672i)5-s − 16.1·7-s + (5.49 + 3.99i)8-s + (−1.20 − 4.64i)10-s + (10.1 + 31.2i)11-s + (−24.0 + 74.0i)13-s + (−2.14 − 6.60i)14-s + (18.4 − 56.6i)16-s + (57.3 + 41.6i)17-s + (80.7 + 58.6i)19-s + (−73.6 + 47.0i)20-s + (−11.4 + 8.29i)22-s + (14.5 + 44.7i)23-s + ⋯ |
| L(s) = 1 | + (0.0469 + 0.144i)2-s + (0.790 − 0.574i)4-s + (−0.998 − 0.0601i)5-s − 0.872·7-s + (0.242 + 0.176i)8-s + (−0.0381 − 0.147i)10-s + (0.278 + 0.856i)11-s + (−0.513 + 1.58i)13-s + (−0.0409 − 0.126i)14-s + (0.287 − 0.885i)16-s + (0.818 + 0.594i)17-s + (0.975 + 0.708i)19-s + (−0.823 + 0.525i)20-s + (−0.110 + 0.0804i)22-s + (0.131 + 0.405i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0574 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0574 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.945828 + 0.892991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.945828 + 0.892991i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (11.1 + 0.672i)T \) |
| good | 2 | \( 1 + (-0.132 - 0.408i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 + 16.1T + 343T^{2} \) |
| 11 | \( 1 + (-10.1 - 31.2i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (24.0 - 74.0i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-57.3 - 41.6i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-80.7 - 58.6i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-14.5 - 44.7i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (16.9 - 12.2i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (120. + 87.8i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (17.1 - 52.7i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (10.6 - 32.8i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 149.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (483. - 351. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (392. - 284. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (91.3 - 281. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-153. - 473. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (419. + 304. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-701. + 509. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-50.3 - 155. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (342. - 248. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (331. + 240. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (353. + 1.08e3i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.12e3 + 819. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−11.96016749645987867823172959164, −11.25775808262186237282621479562, −9.994342374725903162118526220571, −9.337889216199994716198154487095, −7.68893780688319174083836301015, −7.05293803321279573330596647406, −6.03464328824516533102572249238, −4.59137655214769216824467626266, −3.29650753108315762035039668945, −1.57800460386434904796271795385,
0.53211570755378617131619841440, 3.03648595559042159020894417057, 3.42680244869576847442497271909, 5.28210314116489127310338990045, 6.67864606621443156175280674358, 7.54537893323065131539738389697, 8.346399428509303573560535135755, 9.733463549398356716869133274002, 10.83750063924964560031123807149, 11.59026848402348545456326524143
