| L(s) = 1 | + (1.37 + 0.315i)2-s + (−1.89 − 0.786i)3-s + (1.80 + 0.870i)4-s + (2.17 − 1.10i)5-s + (−2.37 − 1.68i)6-s + (−0.151 − 0.247i)7-s + (2.20 + 1.76i)8-s + (0.867 + 0.867i)9-s + (3.34 − 0.841i)10-s + (2.25 + 0.177i)11-s + (−2.73 − 3.06i)12-s + (−0.172 − 0.717i)13-s + (−0.131 − 0.389i)14-s + (−5.00 + 0.393i)15-s + (2.48 + 3.13i)16-s + (−3.59 + 3.07i)17-s + ⋯ |
| L(s) = 1 | + (0.974 + 0.223i)2-s + (−1.09 − 0.454i)3-s + (0.900 + 0.435i)4-s + (0.972 − 0.495i)5-s + (−0.967 − 0.687i)6-s + (−0.0574 − 0.0937i)7-s + (0.780 + 0.625i)8-s + (0.289 + 0.289i)9-s + (1.05 − 0.265i)10-s + (0.680 + 0.0535i)11-s + (−0.789 − 0.886i)12-s + (−0.0477 − 0.198i)13-s + (−0.0350 − 0.104i)14-s + (−1.29 + 0.101i)15-s + (0.621 + 0.783i)16-s + (−0.872 + 0.744i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61369 - 0.148891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61369 - 0.148891i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.37 - 0.315i)T \) |
| 41 | \( 1 + (5.78 + 2.75i)T \) |
| good | 3 | \( 1 + (1.89 + 0.786i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-2.17 + 1.10i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (0.151 + 0.247i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (-2.25 - 0.177i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (0.172 + 0.717i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (3.59 - 3.07i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (3.49 + 0.838i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (2.94 - 2.14i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (3.88 + 3.31i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (2.10 + 6.49i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.56 - 10.9i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-9.67 - 1.53i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (0.192 - 0.314i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (-5.01 + 5.87i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-3.42 - 4.71i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-9.82 + 1.55i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.538 - 6.84i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (-0.568 + 7.22i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (5.96 - 5.96i)T - 73iT^{2} \) |
| 79 | \( 1 + (5.14 - 12.4i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 17.9iT - 83T^{2} \) |
| 89 | \( 1 + (2.99 - 1.83i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (0.792 + 10.0i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−13.01954329746137439246803947720, −11.93474037449141437622854468939, −11.23943727358343709488679778345, −10.05976040287639788939706553605, −8.593197344410761186885198456829, −7.00110473694012262496968190183, −6.12453702105512911552328032592, −5.47908684046747365744606879712, −4.11358394906978712763720091465, −1.89139505385639622185318787972,
2.25496198630676750471297317641, 4.10381220800354999157849317680, 5.32134702591521201991790041515, 6.16088500228455167371950316537, 6.96523452304643700956132877143, 9.155105278434066323187584075149, 10.38553759964191463006053070462, 10.89709563419806883035615058509, 11.86356457546974564657582088740, 12.75785045855559520382323687813
