L(s) = 1 | + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)15-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)20-s + (−0.499 − 0.866i)25-s + 0.999·27-s + (−1 + 1.73i)29-s − 0.999·33-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)15-s + 16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)20-s + (−0.499 − 0.866i)25-s + 0.999·27-s + (−1 + 1.73i)29-s − 0.999·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.167261605\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167261605\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 2 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.617165939713496771234483052794, −8.873277321730015430779078568338, −7.67328203531851811377422707575, −6.96699070314400794099360902047, −6.51989606614620395412128172753, −5.58948473666710481326945648670, −4.59234477865635825934491422907, −3.65357684221529865691539372548, −3.00316903791872139917304019375, −1.69606744992959541903140378407,
0.900154250872804777173279294693, 1.82916672696701128280875576354, 3.11461549321011374022210516117, 4.03672615170658931443114839217, 5.44115461187505276592197455638, 5.87904768139759362611052414295, 6.57620383145966869357765757476, 7.63618700609401585382443780578, 8.034568816534732898264124423411, 8.668346314552491140691315574926