L(s) = 1 | + (0.809 + 0.587i)2-s + (1.65 + 0.500i)3-s + (0.309 + 0.951i)4-s + (0.732 + 1.00i)5-s + (1.04 + 1.37i)6-s + (0.951 − 0.309i)7-s + (−0.309 + 0.951i)8-s + (2.49 + 1.66i)9-s + 1.24i·10-s + (−3.01 − 1.37i)11-s + (0.0358 + 1.73i)12-s + (1.55 − 2.14i)13-s + (0.951 + 0.309i)14-s + (0.709 + 2.03i)15-s + (−0.809 + 0.587i)16-s + (−0.0657 + 0.0477i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (0.957 + 0.289i)3-s + (0.154 + 0.475i)4-s + (0.327 + 0.450i)5-s + (0.427 + 0.563i)6-s + (0.359 − 0.116i)7-s + (−0.109 + 0.336i)8-s + (0.832 + 0.553i)9-s + 0.394i·10-s + (−0.910 − 0.413i)11-s + (0.0103 + 0.499i)12-s + (0.431 − 0.594i)13-s + (0.254 + 0.0825i)14-s + (0.183 + 0.526i)15-s + (−0.202 + 0.146i)16-s + (−0.0159 + 0.0115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.28973 + 1.39699i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.28973 + 1.39699i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (-1.65 - 0.500i)T \) |
| 7 | \( 1 + (-0.951 + 0.309i)T \) |
| 11 | \( 1 + (3.01 + 1.37i)T \) |
good | 5 | \( 1 + (-0.732 - 1.00i)T + (-1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (-1.55 + 2.14i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.0657 - 0.0477i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (6.03 + 1.96i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 4.41iT - 23T^{2} \) |
| 29 | \( 1 + (0.273 + 0.843i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.202 + 0.147i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.01 + 3.12i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.35 - 4.18i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 4.86iT - 43T^{2} \) |
| 47 | \( 1 + (-6.48 - 2.10i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.70 - 2.33i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.184 + 0.0600i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (4.87 + 6.70i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 5.46T + 67T^{2} \) |
| 71 | \( 1 + (7.04 + 9.69i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-15.7 + 5.10i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (5.39 - 7.42i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.69 + 2.68i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 12.0iT - 89T^{2} \) |
| 97 | \( 1 + (-10.4 - 7.61i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.87127451493954963598559055593, −10.53318952070096410621394365944, −9.259833299514811660644123597803, −8.292285495011218071889520966129, −7.69230064046442395877283408630, −6.56898761657841149082367952167, −5.46366379604634187219204315562, −4.37402740126900009685167115322, −3.24898176448525847717168277236, −2.24421400161890133044715496831,
1.64058896519311247793995917812, 2.60263824310779322832357243031, 3.99693330787179443277425861716, 4.89589380020185556080280533549, 6.15935771658553098977460194490, 7.22877024154456166583791981279, 8.362446052805868363896607658291, 8.995470935059514363496659583828, 10.07904892022318608020155667588, 10.82414247568139197239413135502