| L(s) = 1 | + (0.743 + 0.669i)2-s + (−0.817 − 1.52i)3-s + (0.104 + 0.994i)4-s + (2.01 − 0.972i)5-s + (0.414 − 1.68i)6-s + (−0.0644 + 0.0372i)7-s + (−0.587 + 0.809i)8-s + (−1.66 + 2.49i)9-s + (2.14 + 0.624i)10-s + (2.15 − 2.39i)11-s + (1.43 − 0.972i)12-s + (3.68 − 3.31i)13-s + (−0.0728 − 0.0154i)14-s + (−3.13 − 2.27i)15-s + (−0.978 + 0.207i)16-s + (−2.26 + 3.12i)17-s + ⋯ |
| L(s) = 1 | + (0.525 + 0.473i)2-s + (−0.471 − 0.881i)3-s + (0.0522 + 0.497i)4-s + (0.900 − 0.435i)5-s + (0.169 − 0.686i)6-s + (−0.0243 + 0.0140i)7-s + (−0.207 + 0.286i)8-s + (−0.554 + 0.832i)9-s + (0.678 + 0.197i)10-s + (0.649 − 0.721i)11-s + (0.413 − 0.280i)12-s + (1.02 − 0.919i)13-s + (−0.0194 − 0.00413i)14-s + (−0.808 − 0.588i)15-s + (−0.244 + 0.0519i)16-s + (−0.549 + 0.756i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79917 - 0.457123i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79917 - 0.457123i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.743 - 0.669i)T \) |
| 3 | \( 1 + (0.817 + 1.52i)T \) |
| 5 | \( 1 + (-2.01 + 0.972i)T \) |
| good | 7 | \( 1 + (0.0644 - 0.0372i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.15 + 2.39i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-3.68 + 3.31i)T + (1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (2.26 - 3.12i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.452 - 0.328i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.59 + 7.50i)T + (-21.0 - 9.35i)T^{2} \) |
| 29 | \( 1 + (-0.119 - 0.0533i)T + (19.4 + 21.5i)T^{2} \) |
| 31 | \( 1 + (3.89 - 1.73i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-3.01 - 0.979i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-4.71 - 5.23i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (7.42 - 4.28i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.82 + 10.8i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-3.79 - 5.22i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.37 - 4.86i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (9.05 - 10.0i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-2.28 - 5.13i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (9.21 - 6.69i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (12.6 - 4.10i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (7.58 + 3.37i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (3.23 + 0.339i)T + (81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (-0.615 - 1.89i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.74 - 3.92i)T + (-64.9 - 72.0i)T^{2} \) |
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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
−11.12221634614878844960467568772, −10.35674004977438851816848333256, −8.732700835948618223154349391061, −8.407984982715958657413598194781, −7.02227278687365397301501605225, −6.06236224060462209488217210735, −5.76969346234013171368483377387, −4.42689789145236233470144908312, −2.82005771107949958975582704669, −1.23326737264354325685002078753,
1.75949653519102630587366095314, 3.30925155249185299200005803692, 4.30872838736682140341780254475, 5.35624750928055108775899067473, 6.24306135807941277865631330255, 7.07867116484006331909774038982, 9.172717330785374704243924403151, 9.359871817824694878713553742914, 10.37493656063800494729920865267, 11.28550624487978292305276364013
