L(s) = 1 | + (−0.809 − 0.587i)2-s + (−1.31 + 1.12i)3-s + (0.309 + 0.951i)4-s + (−1.88 − 2.60i)5-s + (1.72 − 0.132i)6-s + (−0.951 + 0.309i)7-s + (0.309 − 0.951i)8-s + (0.479 − 2.96i)9-s + 3.21i·10-s + (0.355 + 3.29i)11-s + (−1.47 − 0.907i)12-s + (1.19 − 1.64i)13-s + (0.951 + 0.309i)14-s + (5.41 + 1.30i)15-s + (−0.809 + 0.587i)16-s + (−1.57 + 1.14i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.761 + 0.648i)3-s + (0.154 + 0.475i)4-s + (−0.844 − 1.16i)5-s + (0.705 − 0.0542i)6-s + (−0.359 + 0.116i)7-s + (0.109 − 0.336i)8-s + (0.159 − 0.987i)9-s + 1.01i·10-s + (0.107 + 0.994i)11-s + (−0.425 − 0.261i)12-s + (0.332 − 0.457i)13-s + (0.254 + 0.0825i)14-s + (1.39 + 0.338i)15-s + (−0.202 + 0.146i)16-s + (−0.382 + 0.277i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.422322 + 0.246664i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.422322 + 0.246664i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.31 - 1.12i)T \) |
| 7 | \( 1 + (0.951 - 0.309i)T \) |
| 11 | \( 1 + (-0.355 - 3.29i)T \) |
good | 5 | \( 1 + (1.88 + 2.60i)T + (-1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (-1.19 + 1.64i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.57 - 1.14i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.14 - 0.698i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 6.70iT - 23T^{2} \) |
| 29 | \( 1 + (-0.422 - 1.29i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.07 - 0.779i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.29 - 10.1i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.23 - 3.80i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 7.48iT - 43T^{2} \) |
| 47 | \( 1 + (3.88 + 1.26i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.85 + 5.30i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-11.7 + 3.82i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.27 - 3.13i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 7.77T + 67T^{2} \) |
| 71 | \( 1 + (-3.19 - 4.39i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.22 + 1.37i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-10.0 + 13.8i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (13.6 - 9.93i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 3.34iT - 89T^{2} \) |
| 97 | \( 1 + (8.66 + 6.29i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.42249801035732842525282072714, −10.12002303100565305725470816034, −9.602669951743158553426815310739, −8.667381266354079938445264121593, −7.78015850925635512261542062002, −6.60615268691749002614135802026, −5.27527399270137232469090607100, −4.40811471114655875702318136306, −3.41018297790384144259862058007, −1.19022896441923925965752605718,
0.47293263131551119818163585770, 2.56102573306858163921624091264, 4.02980525048705706549304280447, 5.60830802326071992323412637404, 6.57127026933422906056924926697, 7.04970508337644750646736007061, 7.957656510785249508539982689217, 8.913848113486614850238308464894, 10.28172710430308621292639951783, 10.99079019989649920279143741021