| L(s) = 1 | + (−1.57 − 3.27i)2-s + (−1.75 + 3.63i)3-s + (−5.75 + 7.21i)4-s + (−7.35 − 1.67i)5-s + 14.6·6-s − 2.47i·7-s + (18.5 + 4.23i)8-s + (−4.55 − 5.70i)9-s + (6.10 + 26.7i)10-s + (−9.47 − 11.8i)11-s + (−16.1 − 33.5i)12-s + (−2.97 + 13.0i)13-s + (−8.10 + 3.90i)14-s + (18.9 − 23.8i)15-s + (−7.19 − 31.5i)16-s + (2.29 + 10.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.789 − 1.63i)2-s + (−0.583 + 1.21i)3-s + (−1.43 + 1.80i)4-s + (−1.47 − 0.335i)5-s + 2.44·6-s − 0.353i·7-s + (2.31 + 0.529i)8-s + (−0.505 − 0.634i)9-s + (0.610 + 2.67i)10-s + (−0.861 − 1.07i)11-s + (−1.34 − 2.79i)12-s + (−0.228 + 1.00i)13-s + (−0.578 + 0.278i)14-s + (1.26 − 1.58i)15-s + (−0.449 − 1.96i)16-s + (0.135 + 0.592i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00845443 + 0.0145662i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00845443 + 0.0145662i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (37.2 + 21.5i)T \) |
| good | 2 | \( 1 + (1.57 + 3.27i)T + (-2.49 + 3.12i)T^{2} \) |
| 3 | \( 1 + (1.75 - 3.63i)T + (-5.61 - 7.03i)T^{2} \) |
| 5 | \( 1 + (7.35 + 1.67i)T + (22.5 + 10.8i)T^{2} \) |
| 7 | \( 1 + 2.47iT - 49T^{2} \) |
| 11 | \( 1 + (9.47 + 11.8i)T + (-26.9 + 117. i)T^{2} \) |
| 13 | \( 1 + (2.97 - 13.0i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + (-2.29 - 10.0i)T + (-260. + 125. i)T^{2} \) |
| 19 | \( 1 + (13.0 + 10.3i)T + (80.3 + 351. i)T^{2} \) |
| 23 | \( 1 + (-8.38 - 10.5i)T + (-117. + 515. i)T^{2} \) |
| 29 | \( 1 + (8.57 + 17.8i)T + (-524. + 657. i)T^{2} \) |
| 31 | \( 1 + (-1.08 + 0.522i)T + (599. - 751. i)T^{2} \) |
| 37 | \( 1 + 13.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (45.8 - 22.0i)T + (1.04e3 - 1.31e3i)T^{2} \) |
| 47 | \( 1 + (36.4 - 45.7i)T + (-491. - 2.15e3i)T^{2} \) |
| 53 | \( 1 + (-12.8 - 56.2i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 + (15.5 + 67.9i)T + (-3.13e3 + 1.51e3i)T^{2} \) |
| 61 | \( 1 + (21.1 - 43.8i)T + (-2.32e3 - 2.90e3i)T^{2} \) |
| 67 | \( 1 + (-39.5 + 49.5i)T + (-998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (33.6 + 26.8i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (48.9 + 11.1i)T + (4.80e3 + 2.31e3i)T^{2} \) |
| 79 | \( 1 - 7.79T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-73.8 - 35.5i)T + (4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (-39.8 + 82.8i)T + (-4.93e3 - 6.19e3i)T^{2} \) |
| 97 | \( 1 + (-86.8 - 108. i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−16.43763519369505156916914902820, −15.36522146552103554208176091677, −13.27654325945263386750616910652, −11.89642540714135960764345338095, −11.16626610761798182821730498190, −10.48234787064938621485707836436, −9.119506070822607692267332715286, −7.990863088046024189845462507931, −4.58478936058340100260673436717, −3.57577093732334889519827064206,
0.02484802634116049067458012401, 5.19417320279138089338225945766, 6.79081004954880551823554320111, 7.52571942404544099963947426413, 8.320246147300920544336723979521, 10.29731436214539014352810476567, 11.96231529868344383531717888026, 13.04170057760743300769548091372, 14.88365643128278191974847357673, 15.37281285061866670772290344364
