| L(s) = 1 | + (−0.946 − 1.05i)2-s + (2.51 + 1.03i)3-s + (−0.207 + 1.98i)4-s + (−3.57 + 1.82i)5-s + (−1.28 − 3.62i)6-s + (1.81 + 2.96i)7-s + (2.28 − 1.66i)8-s + (3.09 + 3.09i)9-s + (5.29 + 2.03i)10-s + (−2.33 − 0.183i)11-s + (−2.58 + 4.77i)12-s + (−0.389 − 1.62i)13-s + (1.39 − 4.71i)14-s + (−10.8 + 0.855i)15-s + (−3.91 − 0.825i)16-s + (4.08 − 3.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.669 − 0.742i)2-s + (1.44 + 0.600i)3-s + (−0.103 + 0.994i)4-s + (−1.59 + 0.814i)5-s + (−0.524 − 1.47i)6-s + (0.686 + 1.12i)7-s + (0.808 − 0.588i)8-s + (1.03 + 1.03i)9-s + (1.67 + 0.642i)10-s + (−0.704 − 0.0554i)11-s + (−0.747 + 1.37i)12-s + (−0.108 − 0.450i)13-s + (0.372 − 1.25i)14-s + (−2.80 + 0.220i)15-s + (−0.978 − 0.206i)16-s + (0.991 − 0.847i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.985406 + 0.351290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.985406 + 0.351290i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.946 + 1.05i)T \) |
| 41 | \( 1 + (-0.503 - 6.38i)T \) |
| good | 3 | \( 1 + (-2.51 - 1.03i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (3.57 - 1.82i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-1.81 - 2.96i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (2.33 + 0.183i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (0.389 + 1.62i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-4.08 + 3.49i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-2.93 - 0.704i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (-3.62 + 2.63i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.737 + 0.629i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-0.302 - 0.929i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.36 + 4.20i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (0.136 + 0.0216i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (5.05 - 8.25i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (-0.648 + 0.759i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (3.78 + 5.20i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (8.61 - 1.36i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.0866 - 1.10i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (-0.217 + 2.76i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (2.06 - 2.06i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.924 - 2.23i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 3.03iT - 83T^{2} \) |
| 89 | \( 1 + (-9.51 + 5.83i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (0.578 + 7.35i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−12.67218547459833924079474608158, −11.73423480450764089094421459870, −10.94010835885101348573629280485, −9.844853288205510418274138825029, −8.792523334219122565740780732339, −7.941866993830714225454392616646, −7.55040310789659298231411503673, −4.70223948623992062046976546438, −3.26405142801462855244528647394, −2.74759823137010899020656558628,
1.25011388331739784742353098533, 3.67420577473743526524107832730, 4.91829792247177538352989029782, 7.23798358656372056305002316552, 7.69224038010253776003784245891, 8.254836919143743926188889708396, 9.168492889652003267291331703323, 10.53743603632837721172792175313, 11.74793526038738973207138359545, 13.04640913878622064486595410157
