| L(s) = 1 | + (0.0797 + 1.41i)2-s + (−0.992 + 2.39i)3-s + (−1.98 + 0.225i)4-s + (0.0861 − 0.169i)5-s + (−3.46 − 1.20i)6-s + (0.388 + 1.61i)7-s + (−0.476 − 2.78i)8-s + (−2.63 − 2.63i)9-s + (0.245 + 0.108i)10-s + (0.130 − 0.153i)11-s + (1.43 − 4.98i)12-s + (0.0548 + 0.0895i)13-s + (−2.25 + 0.677i)14-s + (0.319 + 0.374i)15-s + (3.89 − 0.895i)16-s + (0.293 + 3.72i)17-s + ⋯ |
| L(s) = 1 | + (0.0564 + 0.998i)2-s + (−0.572 + 1.38i)3-s + (−0.993 + 0.112i)4-s + (0.0385 − 0.0756i)5-s + (−1.41 − 0.493i)6-s + (0.146 + 0.611i)7-s + (−0.168 − 0.985i)8-s + (−0.877 − 0.877i)9-s + (0.0776 + 0.0342i)10-s + (0.0394 − 0.0461i)11-s + (0.413 − 1.43i)12-s + (0.0152 + 0.0248i)13-s + (−0.602 + 0.181i)14-s + (0.0825 + 0.0966i)15-s + (0.974 − 0.223i)16-s + (0.0711 + 0.903i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0516410 - 0.816229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0516410 - 0.816229i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0797 - 1.41i)T \) |
| 41 | \( 1 + (-3.90 + 5.07i)T \) |
| good | 3 | \( 1 + (0.992 - 2.39i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-0.0861 + 0.169i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-0.388 - 1.61i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (-0.130 + 0.153i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.0548 - 0.0895i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.293 - 3.72i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (3.89 + 2.38i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-4.61 - 3.35i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.459 + 5.83i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (3.12 - 9.62i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.93 - 9.03i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (0.410 + 2.59i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-2.73 + 11.3i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (2.41 + 0.189i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (-5.01 + 6.90i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.823 - 5.20i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-7.00 + 5.98i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (-6.32 - 5.40i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-5.47 + 5.47i)T - 73iT^{2} \) |
| 79 | \( 1 + (4.74 + 1.96i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 4.70iT - 83T^{2} \) |
| 89 | \( 1 + (5.68 - 1.36i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (10.4 - 8.89i)T + (15.1 - 95.8i)T^{2} \) |
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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
−13.50645346469495665235070622769, −12.43686780951404536622288760250, −11.16675448565972001726747684352, −10.21595596805559932060526467100, −9.180915232540032040929786335195, −8.463672318230159560831561418992, −6.83385584149740506802629058018, −5.60552982970679578923276443282, −4.87748667097587929456700914194, −3.66563173913428277565871848144,
0.877472997421568828641173301694, 2.47471112152677605950947491366, 4.36819056611184922487937110077, 5.82988326110633534119048941491, 7.06707586536185955776639183460, 8.103388178420549200846448523429, 9.416168781533921830804142808553, 10.78693192428839452761446509455, 11.31560906829687487719501274387, 12.61725819416260568396610710893
