| L(s) = 1 | + (2.49 + 5.17i)2-s + (−7.43 + 0.557i)3-s + (−10.5 + 13.2i)4-s + (9.68 + 31.4i)5-s + (−21.4 − 37.0i)6-s + (−66.6 − 38.4i)7-s + (−5.54 − 1.26i)8-s + (−25.1 + 3.78i)9-s + (−138. + 128. i)10-s + (42.0 + 52.7i)11-s + (71.3 − 104. i)12-s + (46.0 + 42.7i)13-s + (33.0 − 440. i)14-s + (−89.5 − 228. i)15-s + (53.2 + 233. i)16-s + (364. + 112. i)17-s + ⋯ |
| L(s) = 1 | + (0.623 + 1.29i)2-s + (−0.826 + 0.0619i)3-s + (−0.662 + 0.830i)4-s + (0.387 + 1.25i)5-s + (−0.594 − 1.03i)6-s + (−1.36 − 0.785i)7-s + (−0.0866 − 0.0197i)8-s + (−0.309 + 0.0467i)9-s + (−1.38 + 1.28i)10-s + (0.347 + 0.435i)11-s + (0.495 − 0.726i)12-s + (0.272 + 0.252i)13-s + (0.168 − 2.24i)14-s + (−0.397 − 1.01i)15-s + (0.207 + 0.910i)16-s + (1.25 + 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0292i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0190019 + 1.29799i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0190019 + 1.29799i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-538. + 1.76e3i)T \) |
| good | 2 | \( 1 + (-2.49 - 5.17i)T + (-9.97 + 12.5i)T^{2} \) |
| 3 | \( 1 + (7.43 - 0.557i)T + (80.0 - 12.0i)T^{2} \) |
| 5 | \( 1 + (-9.68 - 31.4i)T + (-516. + 352. i)T^{2} \) |
| 7 | \( 1 + (66.6 + 38.4i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-42.0 - 52.7i)T + (-3.25e3 + 1.42e4i)T^{2} \) |
| 13 | \( 1 + (-46.0 - 42.7i)T + (2.13e3 + 2.84e4i)T^{2} \) |
| 17 | \( 1 + (-364. - 112. i)T + (6.90e4 + 4.70e4i)T^{2} \) |
| 19 | \( 1 + (36.5 - 242. i)T + (-1.24e5 - 3.84e4i)T^{2} \) |
| 23 | \( 1 + (190. - 485. i)T + (-2.05e5 - 1.90e5i)T^{2} \) |
| 29 | \( 1 + (-1.04e3 - 77.9i)T + (6.99e5 + 1.05e5i)T^{2} \) |
| 31 | \( 1 + (1.41e3 + 962. i)T + (3.37e5 + 8.59e5i)T^{2} \) |
| 37 | \( 1 + (-814. + 470. i)T + (9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-844. + 406. i)T + (1.76e6 - 2.20e6i)T^{2} \) |
| 47 | \( 1 + (217. - 273. i)T + (-1.08e6 - 4.75e6i)T^{2} \) |
| 53 | \( 1 + (2.55e3 - 2.36e3i)T + (5.89e5 - 7.86e6i)T^{2} \) |
| 59 | \( 1 + (319. + 1.40e3i)T + (-1.09e7 + 5.25e6i)T^{2} \) |
| 61 | \( 1 + (1.74e3 + 2.56e3i)T + (-5.05e6 + 1.28e7i)T^{2} \) |
| 67 | \( 1 + (-8.01e3 - 1.20e3i)T + (1.92e7 + 5.93e6i)T^{2} \) |
| 71 | \( 1 + (-676. + 265. i)T + (1.86e7 - 1.72e7i)T^{2} \) |
| 73 | \( 1 + (4.05e3 - 4.37e3i)T + (-2.12e6 - 2.83e7i)T^{2} \) |
| 79 | \( 1 + (-3.18e3 + 5.51e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-295. - 3.93e3i)T + (-4.69e7 + 7.07e6i)T^{2} \) |
| 89 | \( 1 + (6.41e3 - 480. i)T + (6.20e7 - 9.35e6i)T^{2} \) |
| 97 | \( 1 + (-2.03e3 - 2.55e3i)T + (-1.96e7 + 8.63e7i)T^{2} \) |
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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
−15.87865473932709345227933748905, −14.52868476818959816704463372246, −13.91858314939893317562956814612, −12.55710792567798628453387071174, −10.87973008392802210590426939941, −9.889554054720626481152246349821, −7.49238052313901983318145896919, −6.46391518294466381751578857679, −5.80230633551832219019862862629, −3.68574702006922159380640458372,
0.78832088008060179796386586616, 3.05599281154660912571905596896, 5.02214538461822334310349871714, 6.11709979394159934927967796941, 8.861244800442694766586365245407, 9.978512063016389211888939915646, 11.38940390696827163993202154182, 12.50518173634325657368662939136, 12.71604242462607416739598675251, 14.11076885780562581492776942245
