L(s) = 1 | + (0.993 + 0.116i)3-s + (0.973 + 0.230i)9-s + (0.997 − 1.34i)11-s + (−0.238 + 1.35i)17-s + (−0.207 − 1.17i)19-s + (−0.686 − 0.727i)25-s + (0.939 + 0.342i)27-s + (1.14 − 1.21i)33-s + (0.569 + 1.90i)41-s + (−0.569 − 0.0665i)43-s + (−0.993 + 0.116i)49-s + (−0.393 + 1.31i)51-s + (−0.0694 − 1.19i)57-s + (−0.473 − 0.635i)59-s + (0.113 + 0.0268i)67-s + ⋯ |
L(s) = 1 | + (0.993 + 0.116i)3-s + (0.973 + 0.230i)9-s + (0.997 − 1.34i)11-s + (−0.238 + 1.35i)17-s + (−0.207 − 1.17i)19-s + (−0.686 − 0.727i)25-s + (0.939 + 0.342i)27-s + (1.14 − 1.21i)33-s + (0.569 + 1.90i)41-s + (−0.569 − 0.0665i)43-s + (−0.993 + 0.116i)49-s + (−0.393 + 1.31i)51-s + (−0.0694 − 1.19i)57-s + (−0.473 − 0.635i)59-s + (0.113 + 0.0268i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.822011133\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.822011133\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.993 - 0.116i)T \) |
good | 5 | \( 1 + (0.686 + 0.727i)T^{2} \) |
| 7 | \( 1 + (0.993 - 0.116i)T^{2} \) |
| 11 | \( 1 + (-0.997 + 1.34i)T + (-0.286 - 0.957i)T^{2} \) |
| 13 | \( 1 + (0.0581 + 0.998i)T^{2} \) |
| 17 | \( 1 + (0.238 - 1.35i)T + (-0.939 - 0.342i)T^{2} \) |
| 19 | \( 1 + (0.207 + 1.17i)T + (-0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (0.993 + 0.116i)T^{2} \) |
| 29 | \( 1 + (-0.893 + 0.448i)T^{2} \) |
| 31 | \( 1 + (-0.597 + 0.802i)T^{2} \) |
| 37 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 41 | \( 1 + (-0.569 - 1.90i)T + (-0.835 + 0.549i)T^{2} \) |
| 43 | \( 1 + (0.569 + 0.0665i)T + (0.973 + 0.230i)T^{2} \) |
| 47 | \( 1 + (-0.597 - 0.802i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.473 + 0.635i)T + (-0.286 + 0.957i)T^{2} \) |
| 61 | \( 1 + (-0.396 + 0.918i)T^{2} \) |
| 67 | \( 1 + (-0.113 - 0.0268i)T + (0.893 + 0.448i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-1.49 + 1.25i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (0.835 + 0.549i)T^{2} \) |
| 83 | \( 1 + (0.539 - 1.80i)T + (-0.835 - 0.549i)T^{2} \) |
| 89 | \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (-0.707 - 1.64i)T + (-0.686 + 0.727i)T^{2} \) |
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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
−9.035597674194945010509135879331, −8.256972879425790895066015043061, −7.88325795205166540152750879113, −6.52165444064488284435541143881, −6.32926534002660771056654654986, −4.95898288902841114599749437418, −4.04576380311407486056518544202, −3.42500123181355991277787349569, −2.44721960608617232424579156293, −1.29864136565821150952511504031,
1.53272300742316856818235962333, 2.29735761717636344661256588093, 3.46968098279942649685091779056, 4.15670295893799546745709338262, 4.99146243533184828163945223420, 6.15158833381975292993276058196, 7.15787623123410143916090408670, 7.38148354284580610452192812966, 8.383105161920266141181950503294, 9.160623943291090007005933394846