L(s) = 1 | + (−0.366 + 0.633i)2-s + (0.5 + 0.866i)3-s + (0.732 + 1.26i)4-s + (−0.5 + 0.866i)5-s − 0.732·6-s + (−0.866 − 2.5i)7-s − 2.53·8-s + (−0.499 + 0.866i)9-s + (−0.366 − 0.633i)10-s + (1.36 + 2.36i)11-s + (−0.732 + 1.26i)12-s + 5.73·13-s + (1.90 + 0.366i)14-s − 0.999·15-s + (−0.535 + 0.928i)16-s + (−3.36 − 5.83i)17-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.448i)2-s + (0.288 + 0.499i)3-s + (0.366 + 0.633i)4-s + (−0.223 + 0.387i)5-s − 0.298·6-s + (−0.327 − 0.944i)7-s − 0.896·8-s + (−0.166 + 0.288i)9-s + (−0.115 − 0.200i)10-s + (0.411 + 0.713i)11-s + (−0.211 + 0.366i)12-s + 1.58·13-s + (0.508 + 0.0978i)14-s − 0.258·15-s + (−0.133 + 0.232i)16-s + (−0.816 − 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.752416 + 0.662617i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.752416 + 0.662617i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.866 + 2.5i)T \) |
good | 2 | \( 1 + (0.366 - 0.633i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.36 - 2.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 + (3.36 + 5.83i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.23 + 2.13i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.633 + 1.09i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.19T + 29T^{2} \) |
| 31 | \( 1 + (3.23 + 5.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.59 - 6.23i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 + 7.19T + 43T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.19 - 7.26i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.09 + 8.83i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.33 + 2.30i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 + (-2.33 - 4.03i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.69 - 11.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 + (-4.56 + 7.90i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−14.02023374441024965819421090205, −13.18491854728157339603073539450, −11.70675238767127503258581400763, −10.89098960203284352188763419662, −9.577915929829843295969819615190, −8.513907458480985139169851277761, −7.28193823974852025302801968297, −6.49993823231428821026950099325, −4.32094839033757970734898059383, −3.09315354450759599404135906495,
1.60408273106223122148677741851, 3.42465522315660724767131844910, 5.75405956470893578673517064721, 6.50858148979081882075054991466, 8.530060341764745914046822776734, 8.928652244357981869079406491368, 10.48238865047600500967617778295, 11.45649471800004331812478790301, 12.36451433380787406288041599613, 13.39749308982028142947665312816