L(s) = 1 | + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (0.104 + 0.994i)5-s + (−0.104 + 0.994i)9-s + (−0.5 − 0.866i)12-s + (−0.669 + 0.743i)15-s + (0.913 + 0.406i)16-s + (0.104 − 0.994i)20-s + (−1 + 1.73i)23-s + (−0.809 + 0.587i)27-s + (−0.913 + 0.406i)31-s + (0.309 − 0.951i)36-s + (−0.309 − 0.951i)37-s − 45-s + (0.978 − 0.207i)47-s + (0.309 + 0.951i)48-s + ⋯ |
L(s) = 1 | + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (0.104 + 0.994i)5-s + (−0.104 + 0.994i)9-s + (−0.5 − 0.866i)12-s + (−0.669 + 0.743i)15-s + (0.913 + 0.406i)16-s + (0.104 − 0.994i)20-s + (−1 + 1.73i)23-s + (−0.809 + 0.587i)27-s + (−0.913 + 0.406i)31-s + (0.309 − 0.951i)36-s + (−0.309 − 0.951i)37-s − 45-s + (0.978 − 0.207i)47-s + (0.309 + 0.951i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9832405106\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9832405106\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 5 | \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 7 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 13 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 31 | \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 61 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 83 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
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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
−10.23798635397707468950543256969, −9.462975438847406192438140430728, −8.876311868481677839672258240149, −7.922457501152871129786330407692, −7.16774503683869927418924115886, −5.83138800411260401297548223681, −5.11615144442798369969551227209, −3.91891488221525270477614763313, −3.41079661368730100747894974500, −2.07066668989673035872327052712,
0.897428320786826787973931314659, 2.34277045188507119603525639726, 3.68861938348009959087195304882, 4.51060132260213977095644642036, 5.48834226824327514366772477092, 6.51768388583849768727952156672, 7.63017972425821458004172882778, 8.388047248144159511393594420730, 8.809542073501810993817121295644, 9.538626996177761741550904850634