| L(s) = 1 | + (−1.04 + 0.949i)2-s + (−0.956 + 2.31i)3-s + (0.196 − 1.99i)4-s + (−0.300 + 0.152i)5-s + (−1.19 − 3.32i)6-s + (−3.46 + 2.12i)7-s + (1.68 + 2.27i)8-s + (−2.30 − 2.30i)9-s + (0.169 − 0.445i)10-s + (0.267 − 3.40i)11-s + (4.41 + 2.35i)12-s + (−2.06 + 0.495i)13-s + (1.61 − 5.52i)14-s + (−0.0661 − 0.839i)15-s + (−3.92 − 0.782i)16-s + (−0.262 − 0.307i)17-s + ⋯ |
| L(s) = 1 | + (−0.741 + 0.671i)2-s + (−0.552 + 1.33i)3-s + (0.0982 − 0.995i)4-s + (−0.134 + 0.0684i)5-s + (−0.486 − 1.35i)6-s + (−1.31 + 0.803i)7-s + (0.595 + 0.803i)8-s + (−0.766 − 0.766i)9-s + (0.0535 − 0.140i)10-s + (0.0807 − 1.02i)11-s + (1.27 + 0.680i)12-s + (−0.572 + 0.137i)13-s + (0.432 − 1.47i)14-s + (−0.0170 − 0.216i)15-s + (−0.980 − 0.195i)16-s + (−0.0637 − 0.0746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0955170 - 0.314117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0955170 - 0.314117i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.04 - 0.949i)T \) |
| 41 | \( 1 + (-3.05 + 5.62i)T \) |
| good | 3 | \( 1 + (0.956 - 2.31i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (0.300 - 0.152i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (3.46 - 2.12i)T + (3.17 - 6.23i)T^{2} \) |
| 11 | \( 1 + (-0.267 + 3.40i)T + (-10.8 - 1.72i)T^{2} \) |
| 13 | \( 1 + (2.06 - 0.495i)T + (11.5 - 5.90i)T^{2} \) |
| 17 | \( 1 + (0.262 + 0.307i)T + (-2.65 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.0388 + 0.161i)T + (-16.9 - 8.62i)T^{2} \) |
| 23 | \( 1 + (-2.58 + 1.87i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (5.71 - 6.68i)T + (-4.53 - 28.6i)T^{2} \) |
| 31 | \( 1 + (-3.33 - 10.2i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.96 - 6.04i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (6.05 + 0.958i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-5.56 - 3.41i)T + (21.3 + 41.8i)T^{2} \) |
| 53 | \( 1 + (-3.52 - 3.01i)T + (8.29 + 52.3i)T^{2} \) |
| 59 | \( 1 + (-0.573 - 0.788i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.52 + 0.716i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (11.4 - 0.900i)T + (66.1 - 10.4i)T^{2} \) |
| 71 | \( 1 + (8.75 + 0.688i)T + (70.1 + 11.1i)T^{2} \) |
| 73 | \( 1 + (-5.21 + 5.21i)T - 73iT^{2} \) |
| 79 | \( 1 + (-13.9 - 5.79i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 9.27iT - 83T^{2} \) |
| 89 | \( 1 + (-8.67 - 14.1i)T + (-40.4 + 79.2i)T^{2} \) |
| 97 | \( 1 + (7.56 - 0.595i)T + (95.8 - 15.1i)T^{2} \) |
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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
−13.63699083606005316710255964763, −12.17572719165721151814774547667, −11.02481981963750041394364588717, −10.26851639535699880127693641886, −9.312019300237541448193068883522, −8.767735330496465957658243868684, −7.00776019706971791659780964860, −5.90111296342067229545487284200, −5.05624732909842673030280753810, −3.28357952757730586279017946704,
0.40209045687129145302065350477, 2.25012182410702008336860448865, 4.03007370031935065732085862094, 6.23610556888091259241334937121, 7.22320157978360520724874003157, 7.76160496559746086849766430841, 9.492385404729881598420959205652, 10.13233953453812192573653285118, 11.47922930876402943828180551699, 12.21427602349470450432945486567
