| L(s) = 1 | + (0.939 − 0.342i)2-s + (0.291 − 1.65i)3-s + (0.766 − 0.642i)4-s + (1.53 + 1.28i)5-s + (−0.291 − 1.65i)6-s + (1.45 + 2.51i)7-s + (0.500 − 0.866i)8-s + (0.166 + 0.0606i)9-s + (1.88 + 0.684i)10-s + (0.5 − 0.866i)11-s + (−0.840 − 1.45i)12-s + (−0.464 − 2.63i)13-s + (2.22 + 1.86i)14-s + (2.57 − 2.16i)15-s + (0.173 − 0.984i)16-s + (−5.01 + 1.82i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.168 − 0.955i)3-s + (0.383 − 0.321i)4-s + (0.685 + 0.575i)5-s + (−0.119 − 0.675i)6-s + (0.549 + 0.951i)7-s + (0.176 − 0.306i)8-s + (0.0555 + 0.0202i)9-s + (0.594 + 0.216i)10-s + (0.150 − 0.261i)11-s + (−0.242 − 0.420i)12-s + (−0.128 − 0.730i)13-s + (0.595 + 0.499i)14-s + (0.664 − 0.557i)15-s + (0.0434 − 0.246i)16-s + (−1.21 + 0.442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.26933 - 0.889521i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.26933 - 0.889521i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.382 - 4.34i)T \) |
| good | 3 | \( 1 + (-0.291 + 1.65i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (-1.53 - 1.28i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.45 - 2.51i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (0.464 + 2.63i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (5.01 - 1.82i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (1.94 - 1.62i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (4.10 + 1.49i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.86 + 8.42i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.63T + 37T^{2} \) |
| 41 | \( 1 + (-1.09 + 6.22i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-5.28 - 4.43i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-10.3 - 3.75i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (6.56 - 5.51i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (8.72 - 3.17i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (6.59 - 5.53i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-12.5 - 4.55i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-3.47 - 2.91i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.426 + 2.42i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.80 - 10.2i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-3.34 - 5.78i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.0912 - 0.517i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-7.14 + 2.60i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.19549299280386785718251246706, −10.44907142136626444378046021298, −9.309587616764178731028883734653, −8.161275970094329525686523190900, −7.28418671591044378276219321044, −6.10206137898850536498719212283, −5.66397201306653970201145294003, −4.09298521219551491375582754476, −2.48995780439647772416166218922, −1.84332092353842602053835294636,
1.86074894273847818035979457793, 3.65390075986034736914091494133, 4.62844732267452737004149725301, 5.06220991046228427295715622432, 6.62675767700755198447522076700, 7.36301960847473146971877458442, 8.915053055487231017663338222543, 9.337567085172843539046842473456, 10.57619104180616017468900711138, 11.09414294689480805377923611037
