| L(s) = 1 | + (−1.12 − 0.854i)2-s + (1.15 − 2.78i)3-s + (0.538 + 1.92i)4-s + (−0.151 − 0.958i)5-s + (−3.68 + 2.15i)6-s + (4.86 + 0.383i)7-s + (1.04 − 2.63i)8-s + (−4.30 − 4.30i)9-s + (−0.648 + 1.21i)10-s + (−0.728 + 0.174i)11-s + (5.98 + 0.723i)12-s + (−3.24 + 3.80i)13-s + (−5.15 − 4.59i)14-s + (−2.84 − 0.683i)15-s + (−3.42 + 2.07i)16-s + (−1.94 + 3.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.796 − 0.604i)2-s + (0.666 − 1.60i)3-s + (0.269 + 0.963i)4-s + (−0.0679 − 0.428i)5-s + (−1.50 + 0.878i)6-s + (1.84 + 0.144i)7-s + (0.367 − 0.929i)8-s + (−1.43 − 1.43i)9-s + (−0.205 + 0.382i)10-s + (−0.219 + 0.0527i)11-s + (1.72 + 0.208i)12-s + (−0.900 + 1.05i)13-s + (−1.37 − 1.22i)14-s + (−0.735 − 0.176i)15-s + (−0.855 + 0.518i)16-s + (−0.472 + 0.770i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.539069 - 0.893431i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.539069 - 0.893431i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.12 + 0.854i)T \) |
| 41 | \( 1 + (5.42 - 3.39i)T \) |
| good | 3 | \( 1 + (-1.15 + 2.78i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (0.151 + 0.958i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (-4.86 - 0.383i)T + (6.91 + 1.09i)T^{2} \) |
| 11 | \( 1 + (0.728 - 0.174i)T + (9.80 - 4.99i)T^{2} \) |
| 13 | \( 1 + (3.24 - 3.80i)T + (-2.03 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.94 - 3.17i)T + (-7.71 - 15.1i)T^{2} \) |
| 19 | \( 1 + (2.75 - 2.35i)T + (2.97 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-0.323 + 0.996i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.778 - 1.27i)T + (-13.1 + 25.8i)T^{2} \) |
| 31 | \( 1 + (-4.75 + 3.45i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.06 + 0.776i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (3.84 - 1.96i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-8.53 + 0.671i)T + (46.4 - 7.35i)T^{2} \) |
| 53 | \( 1 + (2.36 - 1.44i)T + (24.0 - 47.2i)T^{2} \) |
| 59 | \( 1 + (-1.03 - 0.337i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (7.37 + 3.75i)T + (35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (1.02 - 4.25i)T + (-59.6 - 30.4i)T^{2} \) |
| 71 | \( 1 + (2.62 + 10.9i)T + (-63.2 + 32.2i)T^{2} \) |
| 73 | \( 1 + (-0.557 + 0.557i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.99 - 0.827i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 3.72iT - 83T^{2} \) |
| 89 | \( 1 + (0.600 - 7.63i)T + (-87.9 - 13.9i)T^{2} \) |
| 97 | \( 1 + (0.788 - 3.28i)T + (-86.4 - 44.0i)T^{2} \) |
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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
−12.24436548467981227853148983601, −11.83253169307920812647794462543, −10.68679035500660170354620077097, −9.011034004972976970219044062191, −8.307132524072886442713618850194, −7.72745113370102031056684755073, −6.63099022478140020951414240531, −4.52129545272492275981751900058, −2.33078023774370941623828571363, −1.50441705507501645911460257628,
2.61090679008350412249089956239, 4.67027595340547150333555896185, 5.21898782906182899969486684261, 7.29648605908896297457559500771, 8.271446053396125318694083357405, 8.954329236756390877325809015095, 10.22886721141833115622117513289, 10.68373937967855150083800393906, 11.55836354310948315216361782262, 13.85207023114136934238537911233
