| L(s) = 1 | + (3.23 − 2.35i)2-s + (4.94 − 15.2i)4-s + (−90.1 − 65.5i)5-s + (69.8 − 214. i)7-s + (−19.7 − 60.8i)8-s − 445.·10-s + (−243. + 319. i)11-s + (284. − 206. i)13-s + (−279. − 859. i)14-s + (−207. − 150. i)16-s + (−1.08e3 − 784. i)17-s + (143. + 441. i)19-s + (−1.44e3 + 1.04e3i)20-s + (−37.1 + 1.60e3i)22-s + 2.93e3·23-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−1.61 − 1.17i)5-s + (0.538 − 1.65i)7-s + (−0.109 − 0.336i)8-s − 1.41·10-s + (−0.606 + 0.795i)11-s + (0.466 − 0.338i)13-s + (−0.380 − 1.17i)14-s + (−0.202 − 0.146i)16-s + (−0.906 − 0.658i)17-s + (0.0911 + 0.280i)19-s + (−0.806 + 0.586i)20-s + (−0.0163 + 0.706i)22-s + 1.15·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9488888731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9488888731\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (243. - 319. i)T \) |
| good | 5 | \( 1 + (90.1 + 65.5i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-69.8 + 214. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-284. + 206. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.08e3 + 784. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-143. - 441. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 2.93e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-161. + 497. i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (124. - 90.4i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (207. - 638. i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.03e3 - 3.19e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.69e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (234. + 720. i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-8.68e3 + 6.31e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (5.87e3 - 1.80e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.23e4 - 8.95e3i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + 2.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (4.73e4 + 3.43e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-4.22e3 + 1.30e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-3.20e3 + 2.32e3i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (5.91e4 + 4.29e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 - 8.42e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-7.66e4 + 5.56e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−11.18459291168513957209651155728, −10.30206837812297063806478027291, −8.833136098940867444457547440543, −7.73953385346612651291313989120, −7.09665771586008287378268247728, −4.92127506815782356231551283652, −4.48261520438218790039541115500, −3.45980836909340943454544573325, −1.26443207202691432341672542763, −0.25948967443989835894761800979,
2.53393947960810898164554418056, 3.47013880664995342508454450409, 4.79061368338377628491878493859, 6.06857618224387778531768381684, 7.05829314958604229131185615656, 8.235641597758290058378034209017, 8.721786862443906399171634225970, 10.83739968824800579535883638948, 11.36181625397291600062674649802, 12.06165179955523653944724187407
