L(s) = 1 | + (0.644 + 1.98i)2-s + (−1.77 + 1.29i)3-s + (−1.89 + 1.38i)4-s + (−3.70 − 2.69i)6-s − 0.992·7-s + (−0.587 − 0.427i)8-s + (0.566 − 1.74i)9-s + (0.618 + 1.90i)11-s + (1.59 − 4.91i)12-s + (1.04 − 3.20i)13-s + (−0.639 − 1.96i)14-s + (−0.983 + 3.02i)16-s + (2.34 + 1.70i)17-s + 3.82·18-s + (2.09 + 1.51i)19-s + ⋯ |
L(s) = 1 | + (0.455 + 1.40i)2-s + (−1.02 + 0.746i)3-s + (−0.949 + 0.690i)4-s + (−1.51 − 1.10i)6-s − 0.375·7-s + (−0.207 − 0.150i)8-s + (0.188 − 0.581i)9-s + (0.186 + 0.573i)11-s + (0.460 − 1.41i)12-s + (0.289 − 0.889i)13-s + (−0.170 − 0.526i)14-s + (−0.245 + 0.756i)16-s + (0.567 + 0.412i)17-s + 0.901·18-s + (0.479 + 0.348i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.106157 + 0.940684i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.106157 + 0.940684i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-0.644 - 1.98i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (1.77 - 1.29i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 0.992T + 7T^{2} \) |
| 11 | \( 1 + (-0.618 - 1.90i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.04 + 3.20i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.34 - 1.70i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.09 - 1.51i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.40 - 4.32i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.35 + 3.16i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.110 + 0.0802i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.664 + 2.04i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.66 + 8.21i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 4.64T + 43T^{2} \) |
| 47 | \( 1 + (-8.03 + 5.83i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-6.12 + 4.44i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.51 + 4.67i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.855 + 2.63i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.76 - 1.28i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (7.80 - 5.66i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.239 + 0.737i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (12.8 - 9.31i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.43 + 1.04i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-4.48 - 13.7i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (13.7 - 10.0i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.10959918910686807605753565375, −13.04104771635002301526338403574, −11.90983684761856381311757061035, −10.68009400425123140966934183705, −9.778178490966372419216154112622, −8.223344224045782344009580613699, −7.06907241426257062043353459666, −5.85616938902760109613537877075, −5.25385183935734417385814863234, −3.93265429496862221039428894318,
1.12639188219607028131991126692, 3.04573600228044058396216539525, 4.65293761762137282576417576334, 6.07899320678413094032656372611, 7.16014306199789855358376156182, 9.024309862091834607697118087478, 10.25669145670302876882142343774, 11.29418684727369131528264229010, 11.82117192301370845144838396486, 12.63295413301885882287911675742