| L(s) = 1 | + (−10.4 + 4.42i)2-s + (44.2 − 18.3i)3-s + (88.7 − 92.1i)4-s + (128. − 309. i)5-s + (−379. + 386. i)6-s + (−335. − 335. i)7-s + (−516. + 1.35e3i)8-s + (75.4 − 75.4i)9-s + (35.4 + 3.78e3i)10-s + (−2.54e3 − 1.05e3i)11-s + (2.23e3 − 5.70e3i)12-s + (−1.64e3 − 3.98e3i)13-s + (4.98e3 + 2.00e3i)14-s − 1.60e4i·15-s + (−613. − 1.63e4i)16-s − 6.71e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.920 + 0.391i)2-s + (0.946 − 0.391i)3-s + (0.693 − 0.720i)4-s + (0.458 − 1.10i)5-s + (−0.717 + 0.730i)6-s + (−0.369 − 0.369i)7-s + (−0.356 + 0.934i)8-s + (0.0344 − 0.0344i)9-s + (0.0112 + 1.19i)10-s + (−0.576 − 0.238i)11-s + (0.374 − 0.953i)12-s + (−0.208 − 0.502i)13-s + (0.485 + 0.195i)14-s − 1.22i·15-s + (−0.0374 − 0.999i)16-s − 0.331i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.842721 - 0.979163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.842721 - 0.979163i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (10.4 - 4.42i)T \) |
| good | 3 | \( 1 + (-44.2 + 18.3i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-128. + 309. i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (335. + 335. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (2.54e3 + 1.05e3i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (1.64e3 + 3.98e3i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 6.71e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (7.85e3 + 1.89e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-5.12e4 + 5.12e4i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (1.12e5 - 4.64e4i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 9.02e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.13e5 + 2.75e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-5.75e5 + 5.75e5i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (7.42e5 + 3.07e5i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 8.51e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.83e6 - 7.58e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (9.88e5 - 2.38e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-3.55e5 + 1.47e5i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-1.93e6 + 8.00e5i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-3.43e6 - 3.43e6i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (-1.27e6 + 1.27e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 - 5.88e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-1.57e5 - 3.80e5i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-7.40e6 - 7.40e6i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 + 7.41e6T + 8.07e13T^{2} \) |
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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
−15.06021509198501806704258899414, −13.72884423562054417427577782360, −12.75266141348777861498327612481, −10.75047160809383511672359416205, −9.292868726833132687498233362574, −8.493966483001022000572609150200, −7.25609597694626308583848491722, −5.36704436237916445227180506168, −2.47682097716449047300530782285, −0.70772799333740809378605263928,
2.27910804231061827681711756452, 3.35925220263345028063302026429, 6.47880845243018243186945897099, 7.985351023185640081676666528568, 9.371168954616416386995957310682, 10.15621154647754776973617177231, 11.51201535147328649793270916576, 13.16736381287553989448460883960, 14.66493439602753667074624060047, 15.44525959201143434198385317339
