| L(s) = 1 | + (−1.23 + 3.80i)2-s − 12.1i·3-s + (−12.9 − 9.40i)4-s + (13.1 + 9.57i)5-s + (46.0 + 14.9i)6-s + (−42.6 + 13.8i)7-s + (51.7 − 37.6i)8-s + 96.5·9-s + (−52.7 + 38.2i)10-s + (−57.6 − 79.3i)11-s + (−113. + 156. i)12-s + (−360. − 117. i)13-s − 179. i·14-s + (115. − 159. i)15-s + (79.1 + 243. i)16-s + (−1.24e3 − 1.71e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s − 0.776i·3-s + (−0.404 − 0.293i)4-s + (0.235 + 0.171i)5-s + (0.522 + 0.169i)6-s + (−0.329 + 0.106i)7-s + (0.286 − 0.207i)8-s + 0.397·9-s + (−0.166 + 0.121i)10-s + (−0.143 − 0.197i)11-s + (−0.228 + 0.314i)12-s + (−0.591 − 0.192i)13-s − 0.244i·14-s + (0.132 − 0.183i)15-s + (0.0772 + 0.237i)16-s + (−1.04 − 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.438 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.391373 - 0.626550i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.391373 - 0.626550i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 41 | \( 1 + (6.70e3 + 8.42e3i)T \) |
| good | 3 | \( 1 + 12.1iT - 243T^{2} \) |
| 5 | \( 1 + (-13.1 - 9.57i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (42.6 - 13.8i)T + (1.35e4 - 9.87e3i)T^{2} \) |
| 11 | \( 1 + (57.6 + 79.3i)T + (-4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (360. + 117. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.24e3 + 1.71e3i)T + (-4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (2.30e3 - 749. i)T + (2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-587. + 1.80e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (2.88e3 - 3.97e3i)T + (-6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-6.50e3 + 4.72e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (3.01e3 + 2.18e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 43 | \( 1 + (2.40e3 - 7.40e3i)T + (-1.18e8 - 8.64e7i)T^{2} \) |
| 47 | \( 1 + (-2.81e3 - 913. i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (7.69e3 - 1.05e4i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (2.94e3 - 9.05e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.12e4 - 3.46e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-4.15e4 + 5.71e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (6.22e3 + 8.56e3i)T + (-5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 - 2.63e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.35e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 9.28e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (5.50e4 - 1.78e4i)T + (4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (2.13e4 - 2.94e4i)T + (-2.65e9 - 8.16e9i)T^{2} \) |
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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
−13.13772762844182987276336748843, −12.21172281068700565641618199603, −10.64196802749106164359492640029, −9.493254487717529875920340853137, −8.230356042143496938486473093768, −7.03894415005693370172910138364, −6.26233431246536459117589940750, −4.60074737900591391502359781520, −2.31185258529666329863597879605, −0.31919658854051742105858569548,
1.92662010863954567274854426293, 3.77411133371551498354872554341, 4.86187354786356903169304550629, 6.69894707890944004973313802257, 8.409386066475378664544999340818, 9.539840537434306787026693810556, 10.30511751625685128907199501594, 11.27530408033490036136951224848, 12.72400361878241184319965268031, 13.33903191961430211087474664522
