L(s) = 1 | + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (−36.7 + 21.2i)5-s + (−194.5 + 336. i)7-s − 181. i·8-s + 240·10-s + (−1.80e3 − 1.03e3i)11-s + (−707.5 − 1.22e3i)13-s + (1.90e3 − 1.10e3i)14-s + (−512. + 886. i)16-s + 2.36e3i·17-s − 3.06e3·19-s + (−1.17e3 − 678. i)20-s + (5.87e3 + 1.01e4i)22-s + (1.81e4 − 1.04e4i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.293 + 0.169i)5-s + (−0.567 + 0.982i)7-s − 0.353i·8-s + 0.239·10-s + (−1.35 − 0.780i)11-s + (−0.322 − 0.557i)13-s + (0.694 − 0.400i)14-s + (−0.125 + 0.216i)16-s + 0.481i·17-s − 0.447·19-s + (−0.146 − 0.0848i)20-s + (0.552 + 0.956i)22-s + (1.48 − 0.860i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8158376775\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8158376775\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (4.89 + 2.82i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (36.7 - 21.2i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (194.5 - 336. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.80e3 + 1.03e3i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (707.5 + 1.22e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 2.36e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.06e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.81e4 + 1.04e4i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.12e4 + 6.47e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-5.66e3 - 9.81e3i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 4.71e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-5.36e3 + 3.09e3i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (7.25e4 - 1.25e5i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-1.56e5 - 9.00e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + 2.65e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-3.13e5 + 1.80e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.75e5 + 3.03e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (6.01e4 + 1.04e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 3.35e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.75e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-1.26e5 + 2.18e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (2.93e5 + 1.69e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 7.89e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-6.48e5 + 1.12e6i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−11.41253530625185504582006100685, −10.68330085576722190448708848950, −9.617658858514386585348469747712, −8.561249377785672273085305608500, −7.76536008061516214933571673696, −6.35889275018337499515132976036, −5.17834425596388211360321444792, −3.28166674042004454450124813041, −2.42377009916850530940366122988, −0.46420448697845816662695662212,
0.69574310441265592074250664980, 2.46044921420821258752194530509, 4.17198463795152821356581411196, 5.42230522924932487145231378923, 7.04708889338972959874146853795, 7.45311202219600358563901303880, 8.786479613456242535224883177363, 9.905302035467385050084341952053, 10.55155468689137042780586893841, 11.73614402369669427370901555786