L(s) = 1 | + 90.5i·2-s + (385. − 2.15e3i)3-s − 8.19e3·4-s + 1.05e5i·5-s + (1.94e5 + 3.49e4i)6-s − 1.21e6·7-s − 7.41e5i·8-s + (−4.48e6 − 1.66e6i)9-s − 9.52e6·10-s + 1.71e7i·11-s + (−3.16e6 + 1.76e7i)12-s − 6.76e7·13-s − 1.10e8i·14-s + (2.26e8 + 4.06e7i)15-s + 6.71e7·16-s + 1.67e8i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.176 − 0.984i)3-s − 0.500·4-s + 1.34i·5-s + (0.696 + 0.124i)6-s − 1.47·7-s − 0.353i·8-s + (−0.937 − 0.347i)9-s − 0.952·10-s + 0.878i·11-s + (−0.0882 + 0.492i)12-s − 1.07·13-s − 1.04i·14-s + (1.32 + 0.237i)15-s + 0.250·16-s + 0.408i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.176i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(0.0485258 + 0.545860i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0485258 + 0.545860i\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 90.5iT \) |
| 3 | \( 1 + (-385. + 2.15e3i)T \) |
good | 5 | \( 1 - 1.05e5iT - 6.10e9T^{2} \) |
| 7 | \( 1 + 1.21e6T + 6.78e11T^{2} \) |
| 11 | \( 1 - 1.71e7iT - 3.79e14T^{2} \) |
| 13 | \( 1 + 6.76e7T + 3.93e15T^{2} \) |
| 17 | \( 1 - 1.67e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 8.08e8T + 7.99e17T^{2} \) |
| 23 | \( 1 + 1.02e9iT - 1.15e19T^{2} \) |
| 29 | \( 1 + 1.38e10iT - 2.97e20T^{2} \) |
| 31 | \( 1 - 3.15e10T + 7.56e20T^{2} \) |
| 37 | \( 1 + 1.41e11T + 9.01e21T^{2} \) |
| 41 | \( 1 - 7.34e10iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 1.15e11T + 7.38e22T^{2} \) |
| 47 | \( 1 - 6.63e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 - 1.04e12iT - 1.37e24T^{2} \) |
| 59 | \( 1 + 2.50e11iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 5.74e12T + 9.87e24T^{2} \) |
| 67 | \( 1 - 4.64e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 4.07e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 4.29e12T + 1.22e26T^{2} \) |
| 79 | \( 1 + 1.78e13T + 3.68e26T^{2} \) |
| 83 | \( 1 - 5.35e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 + 6.15e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 - 1.58e12T + 6.52e27T^{2} \) |
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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
−19.61545683337725517870678525818, −18.66535615593640327990938674099, −17.32395764857866381848401065896, −15.31396896184992671520771819989, −13.96589727846702875725784211184, −12.40353397594361885499732150717, −9.840631736238301368585436075544, −7.35713014484031204617586525010, −6.40564101039346302257844246107, −2.87767834652859316154557188946,
0.27283648421792362892319342421, 3.26918754319257457606715968709, 5.14067973023852954548439336995, 8.898499919286703670710916589200, 9.957376376338265255475525830888, 12.10351124382580211852189076219, 13.62456085068350837984457795917, 15.92405764183567679224416425982, 16.88222922564243206264219469883, 19.39845821353700139997847020356