| L(s) = 1 | + (−0.281 + 0.959i)2-s + (0.583 − 1.63i)3-s + (−0.841 − 0.540i)4-s + (0.140 − 0.980i)5-s + (1.40 + 1.01i)6-s + (1.30 − 0.594i)7-s + (0.755 − 0.654i)8-s + (−2.31 − 1.90i)9-s + (0.901 + 0.411i)10-s + (3.75 − 1.10i)11-s + (−1.37 + 1.05i)12-s + (−0.318 + 0.698i)13-s + (0.203 + 1.41i)14-s + (−1.51 − 0.801i)15-s + (0.415 + 0.909i)16-s + (0.408 − 0.262i)17-s + ⋯ |
| L(s) = 1 | + (−0.199 + 0.678i)2-s + (0.336 − 0.941i)3-s + (−0.420 − 0.270i)4-s + (0.0630 − 0.438i)5-s + (0.571 + 0.416i)6-s + (0.492 − 0.224i)7-s + (0.267 − 0.231i)8-s + (−0.773 − 0.634i)9-s + (0.284 + 0.130i)10-s + (1.13 − 0.332i)11-s + (−0.396 + 0.305i)12-s + (−0.0884 + 0.193i)13-s + (0.0544 + 0.378i)14-s + (−0.391 − 0.207i)15-s + (0.103 + 0.227i)16-s + (0.0990 − 0.0636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07852 - 0.221970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07852 - 0.221970i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.281 - 0.959i)T \) |
| 3 | \( 1 + (-0.583 + 1.63i)T \) |
| 23 | \( 1 + (1.71 - 4.47i)T \) |
| good | 5 | \( 1 + (-0.140 + 0.980i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.30 + 0.594i)T + (4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-3.75 + 1.10i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (0.318 - 0.698i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.408 + 0.262i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (4.07 - 6.34i)T + (-7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (3.13 + 4.88i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.102 - 0.118i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (5.07 - 0.729i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-3.04 - 0.437i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.209 - 0.181i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 6.75iT - 47T^{2} \) |
| 53 | \( 1 + (1.44 + 3.16i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-11.2 - 5.13i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-11.4 + 9.92i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.96 + 6.69i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (4.19 - 14.2i)T + (-59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (1.16 + 0.746i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (6.31 + 2.88i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.592 - 4.11i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-10.0 + 11.6i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.80 - 0.259i)T + (93.0 + 27.3i)T^{2} \) |
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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
−13.25616971357987886255596375610, −12.28438472722671795157531278020, −11.27664973489307393738442411797, −9.659365651715985387271343609071, −8.634201065343425651613364260705, −7.85771675085808669149244844166, −6.70738370848788463601002608372, −5.68196431066902714667346046861, −3.94379403038788913064472177918, −1.51333087085228192715087839957,
2.42118914896065215502791584101, 3.89917553801210254256272871395, 5.03071879014999496134257350812, 6.82544178078926854251954008354, 8.489459021812064801710114083550, 9.136452878647710813876320385092, 10.31559478755339927325406968648, 11.02423954813029739311911756289, 11.97485877820679825585342204454, 13.23032009178517137420529186004
