| 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
−15.37281285061866670772290344364, −14.88365643128278191974847357673, −13.04170057760743300769548091372, −11.96231529868344383531717888026, −10.29731436214539014352810476567, −8.320246147300920544336723979521, −7.52571942404544099963947426413, −6.79081004954880551823554320111, −5.19417320279138089338225945766, −0.02484802634116049067458012401,
3.57577093732334889519827064206, 4.58478936058340100260673436717, 7.990863088046024189845462507931, 9.119506070822607692267332715286, 10.48234787064938621485707836436, 11.16626610761798182821730498190, 11.89642540714135960764345338095, 13.27654325945263386750616910652, 15.36522146552103554208176091677, 16.43763519369505156916914902820
