| L(s) = 1 | + (1.06 − 1.26i)2-s + (−0.958 − 1.44i)3-s + (−0.124 − 0.708i)4-s + (−0.908 − 2.04i)5-s + (−2.83 − 0.318i)6-s + (−1.22 − 0.215i)7-s + (1.82 + 1.05i)8-s + (−1.16 + 2.76i)9-s + (−3.54 − 1.01i)10-s + (3.30 − 1.20i)11-s + (−0.902 + 0.859i)12-s + (0.380 + 0.452i)13-s + (−1.56 + 1.31i)14-s + (−2.07 + 3.26i)15-s + (4.62 − 1.68i)16-s + (4.46 − 2.58i)17-s + ⋯ |
| L(s) = 1 | + (0.749 − 0.893i)2-s + (−0.553 − 0.832i)3-s + (−0.0624 − 0.354i)4-s + (−0.406 − 0.913i)5-s + (−1.15 − 0.129i)6-s + (−0.461 − 0.0814i)7-s + (0.646 + 0.373i)8-s + (−0.387 + 0.921i)9-s + (−1.12 − 0.321i)10-s + (0.996 − 0.362i)11-s + (−0.260 + 0.248i)12-s + (0.105 + 0.125i)13-s + (−0.418 + 0.351i)14-s + (−0.536 + 0.844i)15-s + (1.15 − 0.420i)16-s + (1.08 − 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.658332 - 1.10751i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.658332 - 1.10751i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.958 + 1.44i)T \) |
| 5 | \( 1 + (0.908 + 2.04i)T \) |
| good | 2 | \( 1 + (-1.06 + 1.26i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (1.22 + 0.215i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.30 + 1.20i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.380 - 0.452i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.46 + 2.58i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 - 2.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.59 - 1.16i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-7.21 - 6.05i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.196 - 1.11i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (4.06 - 2.34i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.43 + 3.71i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.739 + 2.03i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (6.14 + 1.08i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 8.21iT - 53T^{2} \) |
| 59 | \( 1 + (12.6 + 4.62i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.38 - 7.85i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.82 - 3.37i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (3.43 + 5.95i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.23 - 0.715i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.970 + 0.814i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.982 - 1.17i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-6.52 + 11.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.74 - 13.0i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−12.39482787519344671842086109049, −12.26275443184036206197519965508, −11.39782463430485407963574210286, −10.14665348288848856494378046883, −8.578501565478625451523686101810, −7.50499921275841981157308402619, −6.03907278978901923892218574148, −4.77884583433867551364137854652, −3.43739684324698133072527688094, −1.43019066611463953515791286390,
3.56251470023328001801284004008, 4.53232425411602640946292860974, 6.10744282930780271903524796966, 6.51704395399022574796418327005, 7.959015846179593930718557938736, 9.729139897855320357725843566254, 10.40668940555805869256422088504, 11.60166242260456784391864534011, 12.54459849906765304007696910554, 14.08838115620795965801177039485
