| L(s) = 1 | + (5.02 + 1.33i)3-s + (−2.40 + 4.16i)5-s + (3.40 + 18.2i)7-s + (23.4 + 13.4i)9-s + (29.6 − 17.1i)11-s − 17.1i·13-s + (−17.6 + 17.6i)15-s + (50.4 + 87.4i)17-s + (−119. − 69.2i)19-s + (−7.29 + 95.9i)21-s + (0.541 + 0.312i)23-s + (50.9 + 88.2i)25-s + (99.5 + 98.8i)27-s + 222. i·29-s + (−247. + 143. i)31-s + ⋯ |
| L(s) = 1 | + (0.966 + 0.257i)3-s + (−0.215 + 0.372i)5-s + (0.183 + 0.982i)7-s + (0.867 + 0.498i)9-s + (0.813 − 0.469i)11-s − 0.366i·13-s + (−0.303 + 0.304i)15-s + (0.720 + 1.24i)17-s + (−1.44 − 0.835i)19-s + (−0.0757 + 0.997i)21-s + (0.00490 + 0.00283i)23-s + (0.407 + 0.705i)25-s + (0.709 + 0.704i)27-s + 1.42i·29-s + (−1.43 + 0.829i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.592429802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.592429802\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-5.02 - 1.33i)T \) |
| 7 | \( 1 + (-3.40 - 18.2i)T \) |
| good | 5 | \( 1 + (2.40 - 4.16i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-29.6 + 17.1i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 17.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-50.4 - 87.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (119. + 69.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-0.541 - 0.312i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 222. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (247. - 143. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-9.19 + 15.9i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 219.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 370.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-143. + 247. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (29.3 - 16.9i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (127. + 220. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (29.2 + 16.8i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (87.9 + 152. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 626. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-189. + 109. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-263. + 456. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 370.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-412. + 714. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.30e3iT - 9.12e5T^{2} \) |
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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
−11.08267170149499296989209920536, −10.50941677665832451405735963457, −9.025833185295159834090969347835, −8.822735317890532142064372231544, −7.69559531299697294770073464164, −6.57852594411612772201376945333, −5.36053149817121316420410204896, −3.95716832355986239743307984772, −3.01330892910555722892458829775, −1.70905426043707125679485196415,
0.872151478894894805454036368396, 2.23659747776250860960897193629, 3.87615635587202735804468890507, 4.43702625895545798829941744082, 6.26901948108472902958162470187, 7.35397822492701329826077480556, 7.952131758295618771036018473042, 9.107193184101829353924378462934, 9.763645021738135680039782147828, 10.85473099644104979632379804037
