| L(s) = 1 | + (−19.7 + 60.8i)2-s + (950. − 690. i)3-s + (−3.31e3 − 2.40e3i)4-s + (2.04e4 + 6.30e4i)5-s + (2.32e4 + 7.15e4i)6-s + (4.71e3 + 3.42e3i)7-s + (2.12e5 − 1.54e5i)8-s + (−6.62e4 + 2.03e5i)9-s − 4.24e6·10-s + (1.09e6 − 5.77e6i)11-s − 4.81e6·12-s + (−3.50e6 + 1.07e7i)13-s + (−3.01e5 + 2.19e5i)14-s + (6.30e7 + 4.57e7i)15-s + (5.18e6 + 1.59e7i)16-s + (2.71e7 + 8.35e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (0.752 − 0.546i)3-s + (−0.404 − 0.293i)4-s + (0.586 + 1.80i)5-s + (0.203 + 0.625i)6-s + (0.0151 + 0.0109i)7-s + (0.286 − 0.207i)8-s + (−0.0415 + 0.127i)9-s − 1.34·10-s + (0.186 − 0.982i)11-s − 0.465·12-s + (−0.201 + 0.619i)13-s + (−0.0107 + 0.00777i)14-s + (1.42 + 1.03i)15-s + (0.0772 + 0.237i)16-s + (0.272 + 0.839i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.696i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.718 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.963982016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963982016\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.7 - 60.8i)T \) |
| 11 | \( 1 + (-1.09e6 + 5.77e6i)T \) |
| good | 3 | \( 1 + (-950. + 690. i)T + (4.92e5 - 1.51e6i)T^{2} \) |
| 5 | \( 1 + (-2.04e4 - 6.30e4i)T + (-9.87e8 + 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-4.71e3 - 3.42e3i)T + (2.99e10 + 9.21e10i)T^{2} \) |
| 13 | \( 1 + (3.50e6 - 1.07e7i)T + (-2.45e14 - 1.78e14i)T^{2} \) |
| 17 | \( 1 + (-2.71e7 - 8.35e7i)T + (-8.01e15 + 5.82e15i)T^{2} \) |
| 19 | \( 1 + (-5.20e7 + 3.78e7i)T + (1.29e16 - 3.99e16i)T^{2} \) |
| 23 | \( 1 + 6.41e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (1.85e9 + 1.34e9i)T + (3.17e18 + 9.75e18i)T^{2} \) |
| 31 | \( 1 + (1.22e9 - 3.75e9i)T + (-1.97e19 - 1.43e19i)T^{2} \) |
| 37 | \( 1 + (-1.34e10 - 9.78e9i)T + (7.52e19 + 2.31e20i)T^{2} \) |
| 41 | \( 1 + (3.89e10 - 2.83e10i)T + (2.85e20 - 8.79e20i)T^{2} \) |
| 43 | \( 1 - 2.13e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (1.67e10 - 1.21e10i)T + (1.68e21 - 5.19e21i)T^{2} \) |
| 53 | \( 1 + (1.28e10 - 3.96e10i)T + (-2.10e22 - 1.53e22i)T^{2} \) |
| 59 | \( 1 + (-3.95e11 - 2.87e11i)T + (3.24e22 + 9.98e22i)T^{2} \) |
| 61 | \( 1 + (8.63e9 + 2.65e10i)T + (-1.30e23 + 9.51e22i)T^{2} \) |
| 67 | \( 1 - 7.12e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (4.38e11 + 1.34e12i)T + (-9.42e23 + 6.84e23i)T^{2} \) |
| 73 | \( 1 + (-2.08e12 - 1.51e12i)T + (5.16e23 + 1.59e24i)T^{2} \) |
| 79 | \( 1 + (-7.02e11 + 2.16e12i)T + (-3.77e24 - 2.74e24i)T^{2} \) |
| 83 | \( 1 + (6.69e11 + 2.05e12i)T + (-7.17e24 + 5.21e24i)T^{2} \) |
| 89 | \( 1 + 2.47e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (1.14e12 - 3.52e12i)T + (-5.44e25 - 3.95e25i)T^{2} \) |
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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
−14.98667289282962356185054569630, −14.19562323592864810143919930320, −13.49888857900957385011263637590, −11.19937975808228984830487219169, −9.903349312027517504263312543166, −8.272379831806463638892907232498, −7.05291827895772111974013531664, −5.99085065659363227134473268533, −3.30595975013576318735420063926, −1.92629424104142376686721557837,
0.64485932396335415369427843467, 2.11379741516309894779478631772, 3.99113325829608934885356080094, 5.26218387803026445161019283470, 8.079230971005841792983428870563, 9.313656787177038900593178723926, 9.816424252908751306368439610194, 12.03455935096608474794007752629, 12.94024134801741926981566247739, 14.22923151693737042122447387738
