L(s) = 1 | + (−0.689 − 0.501i)2-s + (0.309 − 0.951i)3-s + (−0.393 − 1.21i)4-s + (−0.139 + 0.101i)5-s + (−0.689 + 0.501i)6-s + (−0.572 − 1.76i)7-s + (−0.862 + 2.65i)8-s + (−0.809 − 0.587i)9-s + 0.147·10-s + (−3.17 − 0.954i)11-s − 1.27·12-s + (−0.809 − 0.587i)13-s + (−0.488 + 1.50i)14-s + (0.0533 + 0.164i)15-s + (−0.134 + 0.0976i)16-s + (0.443 − 0.322i)17-s + ⋯ |
L(s) = 1 | + (−0.487 − 0.354i)2-s + (0.178 − 0.549i)3-s + (−0.196 − 0.605i)4-s + (−0.0625 + 0.0454i)5-s + (−0.281 + 0.204i)6-s + (−0.216 − 0.665i)7-s + (−0.304 + 0.938i)8-s + (−0.269 − 0.195i)9-s + 0.0465·10-s + (−0.957 − 0.287i)11-s − 0.367·12-s + (−0.224 − 0.163i)13-s + (−0.130 + 0.401i)14-s + (0.0137 + 0.0424i)15-s + (−0.0336 + 0.0244i)16-s + (0.107 − 0.0781i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0732130 + 0.542505i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0732130 + 0.542505i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (3.17 + 0.954i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
good | 2 | \( 1 + (0.689 + 0.501i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (0.139 - 0.101i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.572 + 1.76i)T + (-5.66 + 4.11i)T^{2} \) |
| 17 | \( 1 + (-0.443 + 0.322i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.849 + 2.61i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 + (-1.91 - 5.90i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.83 + 4.96i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (3.34 + 10.3i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.05 - 3.24i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 + (2.04 - 6.30i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (2.87 + 2.09i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (4.02 + 12.3i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.04 + 1.48i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 0.221T + 67T^{2} \) |
| 71 | \( 1 + (-5.52 + 4.01i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.19 + 9.84i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.20 - 1.59i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-8.18 + 5.94i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 7.26T + 89T^{2} \) |
| 97 | \( 1 + (-12.5 - 9.12i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.78671136310961837536836829002, −9.748230061285372935871114882641, −9.013753924073649219525756983956, −7.914442149081173216002905769943, −7.15806106747909832049289842314, −5.88499388448316631431560057544, −4.97517212125712486009733598444, −3.33202122206070753925433857582, −1.96254734537792728337250424398, −0.38315610418072010641204690504,
2.53801852473773518691172048558, 3.70769063574871932498160743359, 4.87787441520769056114452709407, 6.04311511180178941923383278311, 7.28152116550127100704296090094, 8.172619991696436379907288222582, 8.799602108170843087888274229383, 9.806094297760616931817874075697, 10.37185965311669056697156697033, 11.81475374955300927478846866857