| L(s) = 1 | + (0.978 − 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (−2.65 + 2.94i)5-s + (−0.309 − 0.951i)6-s + (−1.47 + 2.19i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (−1.98 + 3.43i)10-s + (−2.60 + 2.05i)11-s + (−0.5 − 0.866i)12-s + (−1.17 + 3.61i)13-s + (−0.988 + 2.45i)14-s + (3.21 + 2.33i)15-s + (0.669 − 0.743i)16-s + (2.16 + 0.460i)17-s + ⋯ |
| L(s) = 1 | + (0.691 − 0.147i)2-s + (−0.0603 − 0.574i)3-s + (0.456 − 0.203i)4-s + (−1.18 + 1.31i)5-s + (−0.126 − 0.388i)6-s + (−0.558 + 0.829i)7-s + (0.286 − 0.207i)8-s + (−0.326 + 0.0693i)9-s + (−0.627 + 1.08i)10-s + (−0.784 + 0.619i)11-s + (−0.144 − 0.250i)12-s + (−0.325 + 1.00i)13-s + (−0.264 + 0.655i)14-s + (0.829 + 0.602i)15-s + (0.167 − 0.185i)16-s + (0.524 + 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0582 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0582 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.798334 + 0.846267i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.798334 + 0.846267i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 + 0.207i)T \) |
| 3 | \( 1 + (0.104 + 0.994i)T \) |
| 7 | \( 1 + (1.47 - 2.19i)T \) |
| 11 | \( 1 + (2.60 - 2.05i)T \) |
| good | 5 | \( 1 + (2.65 - 2.94i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (1.17 - 3.61i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.16 - 0.460i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-5.71 - 2.54i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (4.09 + 7.08i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.00 + 2.18i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.54 - 1.71i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.583 - 5.55i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (6.95 - 5.04i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 + (-5.55 - 2.47i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.97 - 5.52i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (3.89 - 1.73i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (0.0746 - 0.0828i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-1.94 + 3.37i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.89 - 15.0i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.59 - 1.15i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-1.42 + 0.302i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.197 - 0.608i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (4.22 + 7.31i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.576 - 1.77i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.71445399006069183040520139421, −10.56527447661449569810182426681, −9.772658983886774895957819875143, −8.224270462525674544398048608178, −7.39945505112041943670346456645, −6.70264237174638871513989294850, −5.76943757073145825832113744984, −4.37090416406020331212631445534, −3.17160299090206968690927633464, −2.36399918522939446178573532135,
0.56479064438236026565470906743, 3.29837788290463191726083174500, 3.88197431668996920926837682976, 5.11567160897471936572888096029, 5.58656636568856306256602979020, 7.49193701357277026815430764822, 7.76293898583847940977585619150, 8.995303937340324382645098128332, 9.992317616087568282740241836463, 10.98469429446696097813031847859
