L(s) = 1 | + (0.5 + 0.866i)2-s + (1.43 − 0.972i)3-s + (−0.499 + 0.866i)4-s + (−3.72 + 2.15i)5-s + (1.55 + 0.755i)6-s + (−2.62 + 0.301i)7-s − 0.999·8-s + (1.11 − 2.78i)9-s + (−3.72 − 2.15i)10-s + (−0.167 + 3.31i)11-s + (0.125 + 1.72i)12-s + 2.22i·13-s + (−1.57 − 2.12i)14-s + (−3.24 + 6.70i)15-s + (−0.5 − 0.866i)16-s + (−2.17 + 3.76i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.827 − 0.561i)3-s + (−0.249 + 0.433i)4-s + (−1.66 + 0.961i)5-s + (0.636 + 0.308i)6-s + (−0.993 + 0.113i)7-s − 0.353·8-s + (0.370 − 0.929i)9-s + (−1.17 − 0.679i)10-s + (−0.0505 + 0.998i)11-s + (0.0361 + 0.498i)12-s + 0.617i·13-s + (−0.420 − 0.568i)14-s + (−0.838 + 1.73i)15-s + (−0.125 − 0.216i)16-s + (−0.527 + 0.914i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.210915 + 0.918833i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.210915 + 0.918833i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1.43 + 0.972i)T \) |
| 7 | \( 1 + (2.62 - 0.301i)T \) |
| 11 | \( 1 + (0.167 - 3.31i)T \) |
good | 5 | \( 1 + (3.72 - 2.15i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 2.22iT - 13T^{2} \) |
| 17 | \( 1 + (2.17 - 3.76i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.35 - 0.779i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.48 + 0.859i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.52T + 29T^{2} \) |
| 31 | \( 1 + (1.83 - 3.17i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.36 - 5.83i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 4.66iT - 43T^{2} \) |
| 47 | \( 1 + (3.24 - 1.87i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.14 - 1.23i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.19 - 1.84i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.02 - 2.32i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.08 - 3.61i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.07iT - 71T^{2} \) |
| 73 | \( 1 + (-12.7 - 7.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.48 + 4.89i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + (10.8 - 6.24i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 0.541T + 97T^{2} \) |
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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
−11.69485115206287559981898131997, −10.56899390846746381042052722836, −9.383591560095194745649042068297, −8.442829436729525675049927058549, −7.56596423739549076682846131373, −6.95078997307823195809487293734, −6.34282184054794289962506423718, −4.27929124132899745573203199126, −3.67439073087248382252590538861, −2.56890349896520224997406683714,
0.46081916874382802462236573295, 2.87560167536005061755096626667, 3.69137923691906191685287638615, 4.42233573308820393224310342927, 5.54911798080123644990754811665, 7.26890856395107823106115560910, 8.141501291032928787042433671064, 9.006420538436953903694548648113, 9.578738688507744590123599362033, 10.98461051613512143279534472026