L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.657 − 1.60i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−1.05 + 1.37i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−2.13 + 2.10i)9-s − 0.999·10-s + (−1.37 − 2.37i)11-s + (1.71 + 0.231i)12-s + (2.01 − 3.49i)13-s + (0.499 − 0.866i)14-s + (−1.71 − 0.231i)15-s + (−0.5 − 0.866i)16-s − 7.58·17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.379 − 0.925i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.432 + 0.559i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.711 + 0.702i)9-s − 0.316·10-s + (−0.414 − 0.717i)11-s + (0.495 + 0.0669i)12-s + (0.559 − 0.969i)13-s + (0.133 − 0.231i)14-s + (−0.443 − 0.0598i)15-s + (−0.125 − 0.216i)16-s − 1.84·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.154532 + 0.484416i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.154532 + 0.484416i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.657 + 1.60i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 11 | \( 1 + (1.37 + 2.37i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.01 + 3.49i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 + (-0.873 + 1.51i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.67 + 8.10i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.27 - 9.12i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.30T + 37T^{2} \) |
| 41 | \( 1 + (1.51 - 2.63i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.644 - 1.11i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.378 - 0.655i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 7.23T + 53T^{2} \) |
| 59 | \( 1 + (-0.612 + 1.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.16 + 10.6i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.64 - 11.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.550T + 71T^{2} \) |
| 73 | \( 1 + 9.07T + 73T^{2} \) |
| 79 | \( 1 + (3.14 + 5.44i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.18 + 10.7i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 7.02T + 89T^{2} \) |
| 97 | \( 1 + (-7.79 - 13.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.36769074792833283918700252907, −8.901679425695714897363755540127, −8.524433025255262738652129295495, −7.61944969755350840433052017698, −6.40053066236301056784524066984, −5.63221090116424828678250586749, −4.45914585884115818392003196356, −2.85640863516264206578258713863, −1.83200772442889406083912047009, −0.31078450783809097514467536341,
2.12841428475819288349096045225, 3.96983267550546519952688074784, 4.62626484103366144189219218808, 5.79647597796660280981975241524, 6.63568532776518014291675571749, 7.44402725169564144686368073934, 8.860890422319954800191460617201, 9.190204554467631543778330396907, 10.27593016882491192924448128461, 10.94705287398532109140111527984