| L(s) = 1 | + (−2.76 − 5.73i)2-s + (5.87 + 8.61i)3-s + (−15.2 + 19.1i)4-s + (10.9 − 11.7i)5-s + (33.1 − 57.4i)6-s + (63.9 − 36.9i)7-s + (52.8 + 12.0i)8-s + (−10.1 + 25.8i)9-s + (−97.5 − 30.0i)10-s + (−30.2 − 37.9i)11-s + (−254. − 19.1i)12-s + (27.9 − 8.61i)13-s + (−388. − 264. i)14-s + (165. + 24.9i)15-s + (10.5 + 46.0i)16-s + (361. − 334. i)17-s + ⋯ |
| L(s) = 1 | + (−0.690 − 1.43i)2-s + (0.652 + 0.957i)3-s + (−0.955 + 1.19i)4-s + (0.436 − 0.470i)5-s + (0.921 − 1.59i)6-s + (1.30 − 0.753i)7-s + (0.825 + 0.188i)8-s + (−0.125 + 0.319i)9-s + (−0.975 − 0.300i)10-s + (−0.250 − 0.313i)11-s + (−1.77 − 0.132i)12-s + (0.165 − 0.0509i)13-s + (−1.98 − 1.35i)14-s + (0.734 + 0.110i)15-s + (0.0410 + 0.180i)16-s + (1.24 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0116 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0116 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.01798 - 1.00621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01798 - 1.00621i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-316. - 1.82e3i)T \) |
| good | 2 | \( 1 + (2.76 + 5.73i)T + (-9.97 + 12.5i)T^{2} \) |
| 3 | \( 1 + (-5.87 - 8.61i)T + (-29.5 + 75.4i)T^{2} \) |
| 5 | \( 1 + (-10.9 + 11.7i)T + (-46.7 - 623. i)T^{2} \) |
| 7 | \( 1 + (-63.9 + 36.9i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (30.2 + 37.9i)T + (-3.25e3 + 1.42e4i)T^{2} \) |
| 13 | \( 1 + (-27.9 + 8.61i)T + (2.35e4 - 1.60e4i)T^{2} \) |
| 17 | \( 1 + (-361. + 334. i)T + (6.24e3 - 8.32e4i)T^{2} \) |
| 19 | \( 1 + (242. - 95.3i)T + (9.55e4 - 8.86e4i)T^{2} \) |
| 23 | \( 1 + (148. - 22.4i)T + (2.67e5 - 8.24e4i)T^{2} \) |
| 29 | \( 1 + (-141. + 207. i)T + (-2.58e5 - 6.58e5i)T^{2} \) |
| 31 | \( 1 + (107. - 1.44e3i)T + (-9.13e5 - 1.37e5i)T^{2} \) |
| 37 | \( 1 + (1.48e3 + 856. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (1.90e3 - 917. i)T + (1.76e6 - 2.20e6i)T^{2} \) |
| 47 | \( 1 + (2.60e3 - 3.27e3i)T + (-1.08e6 - 4.75e6i)T^{2} \) |
| 53 | \( 1 + (-2.02e3 - 625. i)T + (6.51e6 + 4.44e6i)T^{2} \) |
| 59 | \( 1 + (-338. - 1.48e3i)T + (-1.09e7 + 5.25e6i)T^{2} \) |
| 61 | \( 1 + (1.88e3 - 141. i)T + (1.36e7 - 2.06e6i)T^{2} \) |
| 67 | \( 1 + (1.66e3 + 4.24e3i)T + (-1.47e7 + 1.37e7i)T^{2} \) |
| 71 | \( 1 + (-946. + 6.27e3i)T + (-2.42e7 - 7.49e6i)T^{2} \) |
| 73 | \( 1 + (-1.63e3 - 5.28e3i)T + (-2.34e7 + 1.59e7i)T^{2} \) |
| 79 | \( 1 + (194. + 337. i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-9.82e3 + 6.70e3i)T + (1.73e7 - 4.41e7i)T^{2} \) |
| 89 | \( 1 + (3.23e3 + 4.74e3i)T + (-2.29e7 + 5.84e7i)T^{2} \) |
| 97 | \( 1 + (-2.58e3 - 3.24e3i)T + (-1.96e7 + 8.63e7i)T^{2} \) |
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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
−14.68956678738607005373399292464, −13.71340410232277040874192577520, −12.19940305304931828320293828655, −10.94481610942272579028360375177, −10.10652916962798509314621542990, −9.096045232342561536112040190440, −8.044922594934167750496672788463, −4.81945826094318551914677795261, −3.31989116921895362303814961397, −1.33930780533629203847259453364,
1.95872559184912447196723900510, 5.45236828268755367620893269378, 6.78861817237812579235465209951, 8.042728302376646643945354587944, 8.517625918882924866194270817890, 10.25479978038396883386845223134, 12.14450372685720660265400052029, 13.69236936064604963638806837731, 14.66023143275201436317179528161, 15.18452714738844534464767223308
