| L(s) = 1 | + (−13.5 + 7.79i)3-s + (151. + 87.3i)5-s + (271. − 209. i)7-s + (121.5 − 210. i)9-s + (−92.6 − 160. i)11-s + 3.98e3i·13-s − 2.72e3·15-s + (6.10e3 − 3.52e3i)17-s + (4.06e3 + 2.34e3i)19-s + (−2.03e3 + 4.94e3i)21-s + (−3.32e3 + 5.75e3i)23-s + (7.45e3 + 1.29e4i)25-s + 3.78e3i·27-s + 1.93e4·29-s + (−2.07e4 + 1.19e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 + 0.288i)3-s + (1.21 + 0.699i)5-s + (0.791 − 0.610i)7-s + (0.166 − 0.288i)9-s + (−0.0696 − 0.120i)11-s + 1.81i·13-s − 0.807·15-s + (1.24 − 0.716i)17-s + (0.592 + 0.342i)19-s + (−0.219 + 0.533i)21-s + (−0.273 + 0.472i)23-s + (0.477 + 0.826i)25-s + 0.192i·27-s + 0.794·29-s + (−0.696 + 0.402i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.769181906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.769181906\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (13.5 - 7.79i)T \) |
| 7 | \( 1 + (-271. + 209. i)T \) |
| good | 5 | \( 1 + (-151. - 87.3i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (92.6 + 160. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 - 3.98e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-6.10e3 + 3.52e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-4.06e3 - 2.34e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (3.32e3 - 5.75e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 - 1.93e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (2.07e4 - 1.19e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-1.97e4 + 3.42e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 4.62e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 9.31e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (1.29e5 + 7.47e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-5.09e4 - 8.81e4i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (1.31e5 - 7.61e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.45e5 + 8.42e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-3.05e4 - 5.29e4i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.85e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (7.73e3 - 4.46e3i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-2.21e5 + 3.84e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 5.59e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (1.79e5 + 1.03e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.23e6iT - 8.32e11T^{2} \) |
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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
−10.65439749239403354750108714364, −9.844250994824283837797684854098, −9.160233040032686194459541999359, −7.62241078256447443628426388559, −6.78407272019382790997046934611, −5.79345561090440125769604594329, −4.88240576972229196938271605166, −3.65032224183720518685858852109, −2.13836818826775140770545242736, −1.12829078372361197744093548398,
0.792401737839959956913222812000, 1.65306985887703798532512550833, 2.91231177432675195020297124659, 4.82098147948863119695084896522, 5.54626217360150459662704474493, 6.09050110198714279636898546172, 7.73541051552201040880301103111, 8.370913962867096883445091918849, 9.600153738049999452860201665358, 10.29807404904639636003384840934
