| L(s) = 1 | + (4 − 6.92i)2-s + (−29.4 − 50.9i)3-s + (−31.9 − 55.4i)4-s + (−212. + 367. i)5-s − 470.·6-s + (−30.7 − 906. i)7-s − 511.·8-s + (−635. + 1.10e3i)9-s + (1.69e3 + 2.93e3i)10-s + (−3.94e3 − 6.83e3i)11-s + (−1.88e3 + 3.25e3i)12-s + 6.71e3·13-s + (−6.40e3 − 3.41e3i)14-s + 2.49e4·15-s + (−2.04e3 + 3.54e3i)16-s + (3.53e3 + 6.12e3i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.628 − 1.08i)3-s + (−0.249 − 0.433i)4-s + (−0.758 + 1.31i)5-s − 0.889·6-s + (−0.0338 − 0.999i)7-s − 0.353·8-s + (−0.290 + 0.503i)9-s + (0.536 + 0.929i)10-s + (−0.893 − 1.54i)11-s + (−0.314 + 0.544i)12-s + 0.848·13-s + (−0.623 − 0.332i)14-s + 1.90·15-s + (−0.125 + 0.216i)16-s + (0.174 + 0.302i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0970i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0444101 - 0.912775i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0444101 - 0.912775i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 + 6.92i)T \) |
| 7 | \( 1 + (30.7 + 906. i)T \) |
| good | 3 | \( 1 + (29.4 + 50.9i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (212. - 367. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (3.94e3 + 6.83e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 6.71e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-3.53e3 - 6.12e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.31e4 + 2.27e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (6.03e3 - 1.04e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 3.06e3T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-5.92e4 - 1.02e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.29e5 + 3.98e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 3.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.16e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (4.03e5 - 6.98e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (2.39e5 + 4.15e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (3.37e5 + 5.84e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.83e5 + 4.90e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (5.92e5 + 1.02e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 4.61e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (1.52e6 + 2.63e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-3.45e6 + 5.97e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 9.01e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-3.50e6 + 6.07e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 8.60e6T + 8.07e13T^{2} \) |
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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
−17.87790768685117903771686299442, −15.94351563642383321493148870860, −14.10160993373354506289037714674, −13.08274203133040915323170417340, −11.33704984769806142530351215698, −10.79876603749808895221493977920, −7.67494122920823692412130931918, −6.25387435315482071445096697991, −3.34472152164228802116647640486, −0.59487739469524278191350608687,
4.40976428558819772352916507757, 5.42508936896971281977099613001, 8.122141389288862376412668412101, 9.690509930483760577729952151658, 11.77964278008920406101954159052, 12.85846393508017066843670375201, 15.25496774519774350831308728644, 15.83437363915695426414582956364, 16.74809572424011048627530119167, 18.29112818618552926710539801667
