L(s) = 1 | + (0.978 − 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.704 + 1.58i)5-s + (0.913 − 0.406i)9-s + (0.5 − 0.866i)12-s + (−0.360 + 1.69i)15-s + (−0.104 − 0.994i)16-s + (0.704 + 1.58i)20-s + (−1.33 − 1.48i)25-s + (0.809 − 0.587i)27-s + (−0.104 + 0.994i)31-s + (0.309 − 0.951i)36-s + (0.309 + 0.951i)37-s + 1.73i·45-s + (−1.28 + 1.15i)47-s + (−0.309 − 0.951i)48-s + ⋯ |
L(s) = 1 | + (0.978 − 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.704 + 1.58i)5-s + (0.913 − 0.406i)9-s + (0.5 − 0.866i)12-s + (−0.360 + 1.69i)15-s + (−0.104 − 0.994i)16-s + (0.704 + 1.58i)20-s + (−1.33 − 1.48i)25-s + (0.809 − 0.587i)27-s + (−0.104 + 0.994i)31-s + (0.309 − 0.951i)36-s + (0.309 + 0.951i)37-s + 1.73i·45-s + (−1.28 + 1.15i)47-s + (−0.309 − 0.951i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.480396685\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.480396685\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 5 | \( 1 + (0.704 - 1.58i)T + (-0.669 - 0.743i)T^{2} \) |
| 7 | \( 1 + (0.913 + 0.406i)T^{2} \) |
| 13 | \( 1 + (-0.978 - 0.207i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 31 | \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (1.28 - 1.15i)T + (0.104 - 0.994i)T^{2} \) |
| 53 | \( 1 + (-1.01 + 1.40i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (1.28 + 1.15i)T + (0.104 + 0.994i)T^{2} \) |
| 61 | \( 1 + (-0.978 + 0.207i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (1.01 + 1.40i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.669 - 0.743i)T^{2} \) |
| 83 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \) |
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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
−10.15172652164896401534150844786, −9.422135668436389056292926362672, −8.184961828674672747467925851708, −7.54055935853766281533074288805, −6.73729561409566838757985366511, −6.32402852515182150731497665505, −4.78419842310782499593966543801, −3.44299673438373742566509584337, −2.88676268636855464829215490878, −1.77885772264160560786016341908,
1.58915416865076229256277851131, 2.85055580356793220769264853200, 3.96301064894381405161402936896, 4.46609725396432904383994528660, 5.71102677016205738954374286308, 7.13306357634824360096387707518, 7.76488796511670902609425494724, 8.428934549238316824918662247416, 8.957337692215816139609724143266, 9.809102828620182518111228923556