| L(s) = 1 | + (−0.618 − 1.90i)2-s + (−3.55 + 2.58i)3-s + (−3.23 + 2.35i)4-s + (7.11 + 5.16i)6-s − 32.4·7-s + (6.47 + 4.70i)8-s + (−2.36 + 7.28i)9-s + (−13.6 − 42.0i)11-s + (5.43 − 16.7i)12-s + (6.12 − 18.8i)13-s + (20.0 + 61.7i)14-s + (4.94 − 15.2i)16-s + (66.6 + 48.4i)17-s + 15.3·18-s + (−8.36 − 6.07i)19-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.684 + 0.497i)3-s + (−0.404 + 0.293i)4-s + (0.484 + 0.351i)6-s − 1.75·7-s + (0.286 + 0.207i)8-s + (−0.0876 + 0.269i)9-s + (−0.374 − 1.15i)11-s + (0.130 − 0.402i)12-s + (0.130 − 0.402i)13-s + (0.383 + 1.17i)14-s + (0.0772 − 0.237i)16-s + (0.950 + 0.690i)17-s + 0.200·18-s + (−0.100 − 0.0733i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.704796 - 0.225160i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.704796 - 0.225160i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.618 + 1.90i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (3.55 - 2.58i)T + (8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 + 32.4T + 343T^{2} \) |
| 11 | \( 1 + (13.6 + 42.0i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-6.12 + 18.8i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-66.6 - 48.4i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (8.36 + 6.07i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-60.0 - 184. i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-159. + 115. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-68.1 - 49.5i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-70.9 + 218. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (6.39 - 19.6i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 36.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-343. + 249. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (4.23 - 3.07i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-16.0 + 49.3i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (38.3 + 118. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (630. + 457. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (496. - 360. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-280. - 863. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-500. + 363. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (365. + 265. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-266. - 821. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-528. + 384. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−11.37755819844347510534378054334, −10.45496750084274233011382209255, −9.956054456683847795229593864484, −8.891295420784006731716719903503, −7.70236605198520680870528844102, −6.14573970273366513325054281113, −5.44502140896446581513498920830, −3.77532622898951007233457901354, −2.89656877580499146304409955055, −0.59354052153110169021617290069,
0.71215168876814203824773464286, 2.97008520418953790018561743324, 4.63771419094322345413259645945, 5.99085408628351269508191948769, 6.67076602372122066813912739898, 7.34972605169583641279602122160, 8.873036305684197113589559886172, 9.745575770841926252215023110856, 10.49399815198347711685162424586, 12.10747330002483899002647726529
