L(s) = 1 | + (−0.447 − 1.34i)2-s + (0.195 + 0.980i)3-s + (−1.59 + 1.20i)4-s + (−1.27 + 0.850i)5-s + (1.22 − 0.700i)6-s + (2.34 + 0.971i)7-s + (2.32 + 1.60i)8-s + (−0.923 + 0.382i)9-s + (1.71 + 1.32i)10-s + (1.71 + 0.340i)11-s + (−1.48 − 1.33i)12-s + (5.39 + 3.60i)13-s + (0.254 − 3.57i)14-s + (−1.08 − 1.08i)15-s + (1.11 − 3.84i)16-s + (−2.15 + 2.15i)17-s + ⋯ |
L(s) = 1 | + (−0.316 − 0.948i)2-s + (0.112 + 0.566i)3-s + (−0.799 + 0.600i)4-s + (−0.569 + 0.380i)5-s + (0.501 − 0.285i)6-s + (0.886 + 0.367i)7-s + (0.822 + 0.569i)8-s + (−0.307 + 0.127i)9-s + (0.540 + 0.419i)10-s + (0.515 + 0.102i)11-s + (−0.429 − 0.385i)12-s + (1.49 + 0.999i)13-s + (0.0679 − 0.956i)14-s + (−0.279 − 0.279i)15-s + (0.279 − 0.960i)16-s + (−0.521 + 0.521i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.967235 + 0.127417i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.967235 + 0.127417i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.447 + 1.34i)T \) |
| 3 | \( 1 + (-0.195 - 0.980i)T \) |
good | 5 | \( 1 + (1.27 - 0.850i)T + (1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-2.34 - 0.971i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-1.71 - 0.340i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-5.39 - 3.60i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (2.15 - 2.15i)T - 17iT^{2} \) |
| 19 | \( 1 + (2.04 - 3.06i)T + (-7.27 - 17.5i)T^{2} \) |
| 23 | \( 1 + (2.77 + 6.70i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.94 + 0.386i)T + (26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + 10.7iT - 31T^{2} \) |
| 37 | \( 1 + (-1.02 - 1.52i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (-1.02 - 2.47i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (0.981 - 4.93i)T + (-39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (-4.74 + 4.74i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.14 + 0.825i)T + (48.9 + 20.2i)T^{2} \) |
| 59 | \( 1 + (-0.855 + 0.571i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (0.548 + 2.75i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (1.28 + 6.44i)T + (-61.8 + 25.6i)T^{2} \) |
| 71 | \( 1 + (-7.53 - 3.12i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-14.7 + 6.10i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (5.81 + 5.81i)T + 79iT^{2} \) |
| 83 | \( 1 + (-1.78 + 2.67i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (5.30 - 12.8i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + 11.3iT - 97T^{2} \) |
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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
−12.21780751853010207212022624605, −11.27846582123287128814718378348, −10.96723856081605039965081128473, −9.683612617444056582617406398428, −8.618801721562487973130482848032, −8.048579471216427107834464850092, −6.25621641174509162122149167330, −4.44215622263129141556372777142, −3.77884594278252677606144557883, −1.97663041069596894065833639607,
1.12700736050627438234159649181, 3.89773926829707762270859514288, 5.18754824991455654038738155679, 6.41881463619900765987393719145, 7.52189375732554613301199633279, 8.328246603716061822130214175225, 8.982222672078173310975690336041, 10.57028995291879187643673620521, 11.44832078629312097029444257685, 12.71156119648266600416386394910