L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − i·5-s − 0.999i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 0.999i·18-s + (0.866 − 0.499i)20-s + (0.5 − 0.866i)29-s + (0.866 − 0.499i)32-s + 0.999i·34-s + (0.499 − 0.866i)36-s + (−0.866 − 0.5i)37-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − i·5-s − 0.999i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 0.999i·18-s + (0.866 − 0.499i)20-s + (0.5 − 0.866i)29-s + (0.866 − 0.499i)32-s + 0.999i·34-s + (0.499 − 0.866i)36-s + (−0.866 − 0.5i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5711930066\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5711930066\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + iT - T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
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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
−10.32895554073747545570199048374, −9.380377363190463904985710672692, −8.926379145636046520401617200361, −8.162831434962240569151754772311, −7.11380780006616797039149253379, −6.12952934496471266055220023215, −4.83363210795382071037005506306, −3.69714944855823528783316673315, −2.45626233076909153801889847746, −0.852200475815374476339511689986,
1.95662763632202046635197634862, 3.08695566288795130996984523323, 4.78696513787114465596855014276, 5.88734466128779035642889046141, 6.67732483656634020697300820469, 7.50173042721762220985295565719, 8.330322232210478869438892599588, 9.103693646329961492069687257275, 10.26820207767872637651649169159, 10.74691411300486468608097107938