| L(s) = 1 | + (−5.65 − 9.79i)2-s + (−40.5 − 23.3i)3-s + (−63.9 + 110. i)4-s + (−1.03e3 + 598. i)5-s + 529. i·6-s + (−2.20e3 − 954. i)7-s + 1.44e3·8-s + (1.09e3 + 1.89e3i)9-s + (1.17e4 + 6.76e3i)10-s + (1.47e3 − 2.54e3i)11-s + (5.18e3 − 2.99e3i)12-s + 1.71e4i·13-s + (3.11e3 + 2.69e4i)14-s + 5.59e4·15-s + (−8.19e3 − 1.41e4i)16-s + (−1.05e5 − 6.07e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−1.65 + 0.957i)5-s + 0.408i·6-s + (−0.917 − 0.397i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (1.17 + 0.676i)10-s + (0.100 − 0.174i)11-s + (0.249 − 0.144i)12-s + 0.599i·13-s + (0.0810 + 0.702i)14-s + 1.10·15-s + (−0.125 − 0.216i)16-s + (−1.26 − 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.440572 - 0.187712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.440572 - 0.187712i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.65 + 9.79i)T \) |
| 3 | \( 1 + (40.5 + 23.3i)T \) |
| 7 | \( 1 + (2.20e3 + 954. i)T \) |
| good | 5 | \( 1 + (1.03e3 - 598. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-1.47e3 + 2.54e3i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 1.71e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (1.05e5 + 6.07e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-9.28e4 + 5.36e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-1.34e5 - 2.33e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.22e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (4.68e5 + 2.70e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.01e6 + 1.76e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 3.23e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 6.67e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (3.92e6 - 2.26e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (6.27e6 - 1.08e7i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (4.99e5 + 2.88e5i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-8.77e6 + 5.06e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.18e6 - 2.05e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.95e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.72e7 - 9.97e6i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (6.01e3 + 1.04e4i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 1.41e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-6.90e7 + 3.98e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 1.14e8iT - 7.83e15T^{2} \) |
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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
−13.93897517656121811543025920057, −12.59074071157122877212597673505, −11.41630643459082885089835224674, −10.95847544542958887231920971432, −9.314474138379571187484363224203, −7.55817976918622660137720015872, −6.76735009197173129082597025385, −4.21293283016619080924043187753, −2.95948611547041659776411325713, −0.47025589679282331440627257050,
0.56670642067079263788394506804, 3.80763484773898109554878701665, 5.14635942291108152711376449234, 6.77814989648199780100917536165, 8.198347026972524696890823841403, 9.198233353290256291337228831729, 10.79209372257266625241020720160, 12.13765245341990851837593966096, 12.94364398197470840009280452011, 15.04603331444639445112728458458
