L(s) = 1 | + (12 + 9.94i)3-s + 79.5i·5-s + 179. i·7-s + (45 + 238. i)9-s + 648·11-s + 242·13-s + (−792 + 955. i)15-s + 318. i·17-s − 1.25e3i·19-s + (−1.78e3 + 2.14e3i)21-s − 1.29e3·23-s − 3.21e3·25-s + (−1.83e3 + 3.31e3i)27-s − 1.98e3i·29-s − 3.40e3i·31-s + ⋯ |
L(s) = 1 | + (0.769 + 0.638i)3-s + 1.42i·5-s + 1.38i·7-s + (0.185 + 0.982i)9-s + 1.61·11-s + 0.397·13-s + (−0.908 + 1.09i)15-s + 0.267i·17-s − 0.796i·19-s + (−0.881 + 1.06i)21-s − 0.510·23-s − 1.02·25-s + (−0.484 + 0.874i)27-s − 0.439i·29-s − 0.635i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.821363898\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.821363898\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-12 - 9.94i)T \) |
good | 5 | \( 1 - 79.5iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 179. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 648T + 1.61e5T^{2} \) |
| 13 | \( 1 - 242T + 3.71e5T^{2} \) |
| 17 | \( 1 - 318. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.25e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.98e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 3.40e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.20e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.48e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.80e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.69e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 8.42e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.02e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.57e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.91e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 7.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.42e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−11.70195792194249950649138961505, −11.13980508107359595599901028338, −9.866194319862985254624453985815, −9.157887117672294528940696415866, −8.182251404421079821546200406183, −6.81339465976824762791958469538, −5.87110770408895107730628401812, −4.16756123793512225662955790163, −3.06846339842153010034966697622, −2.08752641392113306644893999713,
0.877705713219135445031799578805, 1.47339594686954135067058355373, 3.65161404112628769626621923557, 4.42965363099521917612615083294, 6.19808371796912536384200007598, 7.27699135984698108832791716690, 8.296900105757629231271298032361, 9.093134977812673500917715571882, 9.992710033388610245655190502357, 11.54940421548692553848546479804