L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 2·5-s + (−2.5 − 0.866i)7-s − 0.999·8-s + (1 + 1.73i)10-s − 5·11-s + (−3 − 5.19i)13-s + (−0.500 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (−2 − 3.46i)17-s + (2 − 3.46i)19-s + (−0.999 + 1.73i)20-s + (−2.5 − 4.33i)22-s + 4·23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + 0.894·5-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (0.316 + 0.547i)10-s − 1.50·11-s + (−0.832 − 1.44i)13-s + (−0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.485 − 0.840i)17-s + (0.458 − 0.794i)19-s + (−0.223 + 0.387i)20-s + (−0.533 − 0.923i)22-s + 0.834·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8905176539\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8905176539\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + (-3.5 + 6.06i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (6.5 + 11.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.5 - 6.06i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.884316790736952334210472307277, −8.758849555526724694732995710228, −7.74957499841394110253826871885, −7.10805063365404444849785627231, −6.21287246415767166041697011142, −5.29138655921383612684097031691, −4.83410425169887711640233829924, −3.10778967977025177919746099015, −2.63216178768300024532900822954, −0.31258898722708403492739524059,
1.85822350076762921359988334248, 2.60806904154625537109081850329, 3.70557314922337205882333470670, 4.94620843311078224530031315697, 5.63275434321299209053747469853, 6.50629627140443526618675609696, 7.38606273101212781084744721201, 8.737261900673905549961494003795, 9.330232691526209378466732584214, 10.19620435109406487379463914666