L(s) = 1 | + (−0.173 + 0.984i)3-s + (−0.939 − 0.342i)9-s + (−0.266 + 1.50i)11-s + (0.939 + 1.62i)17-s + (0.173 − 0.300i)19-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)27-s + (−1.43 − 0.524i)33-s + (−0.326 − 0.118i)41-s + (0.326 − 1.85i)43-s + (0.173 + 0.984i)49-s + (−1.76 + 0.642i)51-s + (0.266 + 0.223i)57-s + (−0.0603 − 0.342i)59-s + (1.43 + 0.524i)67-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)3-s + (−0.939 − 0.342i)9-s + (−0.266 + 1.50i)11-s + (0.939 + 1.62i)17-s + (0.173 − 0.300i)19-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)27-s + (−1.43 − 0.524i)33-s + (−0.326 − 0.118i)41-s + (0.326 − 1.85i)43-s + (0.173 + 0.984i)49-s + (−1.76 + 0.642i)51-s + (0.266 + 0.223i)57-s + (−0.0603 − 0.342i)59-s + (1.43 + 0.524i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8604669266\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8604669266\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.173 - 0.984i)T \) |
good | 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 11 | \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 13 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
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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
−10.33681471956565338999840744755, −9.958519668671083339393124631448, −9.091584351918052383365857170490, −8.132563934465243470008556657694, −7.24873218296335389305051990575, −6.05937441505121674030918278128, −5.25725246553453368970832890388, −4.29690578283899426179963838290, −3.48038191394335792638921780288, −2.01109970149798648382494256556,
0.923867651047348099398639848761, 2.54837736362177468293033411053, 3.47266272589055496995336154381, 5.12943223799191162585901759655, 5.82120811786269020488655950824, 6.68623866776328465730773752713, 7.71482166484025105733971272805, 8.197001831674674982033674726616, 9.221692111433428009375311054032, 10.16120082854218140931947120499