| L(s) = 1 | + (−64 + 110. i)2-s + (−309. − 535. i)3-s + (−8.19e3 − 1.41e4i)4-s + (8.98e3 − 1.55e4i)5-s + 7.91e4·6-s + (−2.17e6 + 7.98e4i)7-s + 2.09e6·8-s + (6.98e6 − 1.20e7i)9-s + (1.15e6 + 1.99e6i)10-s + (1.74e7 + 3.02e7i)11-s + (−5.06e6 + 8.76e6i)12-s + 1.86e8·13-s + (1.30e8 − 2.46e8i)14-s − 1.11e7·15-s + (−1.34e8 + 2.32e8i)16-s + (−4.88e6 − 8.46e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.0815 − 0.141i)3-s + (−0.249 − 0.433i)4-s + (0.0514 − 0.0890i)5-s + 0.115·6-s + (−0.999 + 0.0366i)7-s + 0.353·8-s + (0.486 − 0.842i)9-s + (0.0363 + 0.0629i)10-s + (0.269 + 0.467i)11-s + (−0.0407 + 0.0706i)12-s + 0.825·13-s + (0.330 − 0.624i)14-s − 0.0167·15-s + (−0.125 + 0.216i)16-s + (−0.00288 − 0.00500i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0998 - 0.994i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.0998 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.756317 + 0.836037i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.756317 + 0.836037i\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (64 - 110. i)T \) |
| 7 | \( 1 + (2.17e6 - 7.98e4i)T \) |
| good | 3 | \( 1 + (309. + 535. i)T + (-7.17e6 + 1.24e7i)T^{2} \) |
| 5 | \( 1 + (-8.98e3 + 1.55e4i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 11 | \( 1 + (-1.74e7 - 3.02e7i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 - 1.86e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + (4.88e6 + 8.46e6i)T + (-1.43e18 + 2.47e18i)T^{2} \) |
| 19 | \( 1 + (1.53e9 - 2.65e9i)T + (-7.59e18 - 1.31e19i)T^{2} \) |
| 23 | \( 1 + (1.45e10 - 2.52e10i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 - 5.56e9T + 8.62e21T^{2} \) |
| 31 | \( 1 + (-1.31e11 - 2.28e11i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 + (-9.69e10 + 1.67e11i)T + (-1.66e23 - 2.88e23i)T^{2} \) |
| 41 | \( 1 + 2.63e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 6.20e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + (-2.11e12 + 3.65e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 + (-3.72e12 - 6.45e12i)T + (-3.65e25 + 6.33e25i)T^{2} \) |
| 59 | \( 1 + (-1.38e13 - 2.40e13i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (1.06e13 - 1.84e13i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (-2.33e12 - 4.03e12i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 - 3.38e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + (3.36e12 + 5.82e12i)T + (-4.45e27 + 7.71e27i)T^{2} \) |
| 79 | \( 1 + (-1.12e14 + 1.95e14i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + 2.65e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + (-4.43e13 + 7.69e13i)T + (-8.70e28 - 1.50e29i)T^{2} \) |
| 97 | \( 1 + 9.04e14T + 6.33e29T^{2} \) |
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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
−16.16517875209956926630891355417, −15.18256962042785014384665788425, −13.52758164794634213172782293592, −12.19005089440221704579348808788, −10.13445717558854407588149360472, −8.969155598264934617074814110004, −7.11271524896363268218274694832, −5.93834462592639259422216305685, −3.74366160304695595506412910378, −1.22470326939134989348180117783,
0.53755491935759590624496448185, 2.49649497109478321044126013153, 4.19027265752742897518955774939, 6.41612196192760828273561085823, 8.358389436341204090440098829629, 9.904782172544192588146419142291, 11.01187275646516166145364214239, 12.67093729080960063185770479518, 13.78681899301907191026758090520, 15.84822929756152564508598974989
