L(s) = 1 | + (−0.832 + 1.44i)2-s + (−0.579 + 1.00i)3-s + (−0.385 − 0.667i)4-s + (−0.965 − 1.67i)6-s + 2.43·7-s − 2.04·8-s + (0.827 + 1.43i)9-s − 5.75·11-s + 0.893·12-s + (−0.797 − 1.38i)13-s + (−2.02 + 3.51i)14-s + (2.47 − 4.28i)16-s + (−2.99 + 5.18i)17-s − 2.75·18-s + (0.149 + 4.35i)19-s + ⋯ |
L(s) = 1 | + (−0.588 + 1.01i)2-s + (−0.334 + 0.579i)3-s + (−0.192 − 0.333i)4-s + (−0.394 − 0.682i)6-s + 0.920·7-s − 0.723·8-s + (0.275 + 0.477i)9-s − 1.73·11-s + 0.258·12-s + (−0.221 − 0.383i)13-s + (−0.541 + 0.938i)14-s + (0.618 − 1.07i)16-s + (−0.725 + 1.25i)17-s − 0.649·18-s + (0.0342 + 0.999i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.211714 - 0.521914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.211714 - 0.521914i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-0.149 - 4.35i)T \) |
good | 2 | \( 1 + (0.832 - 1.44i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.579 - 1.00i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 + (0.797 + 1.38i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.99 - 5.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.470 + 0.814i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.30 + 2.26i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 - 2.89T + 37T^{2} \) |
| 41 | \( 1 + (-3.15 + 5.46i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.26 + 3.93i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.47 - 7.75i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.09 + 1.90i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.39 + 9.35i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.26 - 9.11i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.504 - 0.874i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.41 - 7.65i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.12 - 8.87i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.80 - 6.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.11T + 83T^{2} \) |
| 89 | \( 1 + (-5.55 - 9.62i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.02 + 3.51i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.17899199114698021413062968229, −10.58170402805179353459346262338, −9.803092386344888153594438443712, −8.495773705185641376200674157386, −7.954785316602310433947704628783, −7.30819307433730056202549933353, −5.81862575174831443444016050713, −5.30024857304446586812343953342, −4.07715839251525170412492422850, −2.29323508449266073406875359089,
0.40962720218484536868611886743, 1.90074421752288511967960047843, 2.88674073307965261825454948967, 4.64892352457149715638169256068, 5.65570197556066702788076549681, 6.96345853090113631556430199140, 7.75153180958367891387072729062, 8.913112503613673153515157381568, 9.638324526639629460312854217087, 10.69525300022404574417665795087