| L(s) = 1 | + (−64.3 − 111. i)2-s + (8.11e3 − 1.40e4i)4-s + (1.62e5 − 2.81e5i)5-s + (1.60e5 + 2.77e5i)7-s − 6.30e6·8-s − 4.18e7·10-s + (−5.96e7 − 1.03e8i)11-s + (−2.27e7 + 3.93e7i)13-s + (2.05e7 − 3.56e7i)14-s + (1.39e8 + 2.41e8i)16-s + 9.87e8·17-s + 3.70e9·19-s + (−2.64e9 − 4.57e9i)20-s + (−7.67e9 + 1.32e10i)22-s + (6.91e9 − 1.19e10i)23-s + ⋯ |
| L(s) = 1 | + (−0.355 − 0.615i)2-s + (0.247 − 0.428i)4-s + (0.931 − 1.61i)5-s + (0.0734 + 0.127i)7-s − 1.06·8-s − 1.32·10-s + (−0.923 − 1.59i)11-s + (−0.100 + 0.173i)13-s + (0.0522 − 0.0904i)14-s + (0.129 + 0.224i)16-s + 0.583·17-s + 0.951·19-s + (−0.461 − 0.798i)20-s + (−0.655 + 1.13i)22-s + (0.423 − 0.733i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.667159433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.667159433\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (64.3 + 111. i)T + (-1.63e4 + 2.83e4i)T^{2} \) |
| 5 | \( 1 + (-1.62e5 + 2.81e5i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 7 | \( 1 + (-1.60e5 - 2.77e5i)T + (-2.37e12 + 4.11e12i)T^{2} \) |
| 11 | \( 1 + (5.96e7 + 1.03e8i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 + (2.27e7 - 3.93e7i)T + (-2.55e16 - 4.43e16i)T^{2} \) |
| 17 | \( 1 - 9.87e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 3.70e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + (-6.91e9 + 1.19e10i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 + (-2.42e10 - 4.20e10i)T + (-4.31e21 + 7.47e21i)T^{2} \) |
| 31 | \( 1 + (-2.04e10 + 3.54e10i)T + (-1.17e22 - 2.03e22i)T^{2} \) |
| 37 | \( 1 + 1.69e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + (6.53e11 - 1.13e12i)T + (-7.77e23 - 1.34e24i)T^{2} \) |
| 43 | \( 1 + (-9.12e10 - 1.58e11i)T + (-1.58e24 + 2.75e24i)T^{2} \) |
| 47 | \( 1 + (-2.68e12 - 4.64e12i)T + (-6.03e24 + 1.04e25i)T^{2} \) |
| 53 | \( 1 + 1.72e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + (-1.05e12 + 1.82e12i)T + (-1.82e26 - 3.16e26i)T^{2} \) |
| 61 | \( 1 + (6.15e11 + 1.06e12i)T + (-3.01e26 + 5.21e26i)T^{2} \) |
| 67 | \( 1 + (1.97e13 - 3.42e13i)T + (-1.23e27 - 2.13e27i)T^{2} \) |
| 71 | \( 1 + 4.85e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 7.75e12T + 8.90e27T^{2} \) |
| 79 | \( 1 + (-3.67e13 - 6.36e13i)T + (-1.45e28 + 2.52e28i)T^{2} \) |
| 83 | \( 1 + (4.29e13 + 7.43e13i)T + (-3.05e28 + 5.29e28i)T^{2} \) |
| 89 | \( 1 - 3.61e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + (1.71e14 + 2.96e14i)T + (-3.16e29 + 5.48e29i)T^{2} \) |
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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
−13.03007487256255010313108333562, −11.83411658916353418844032905931, −10.45808180875803129554929884831, −9.323556747875316829850443792087, −8.379732996572334580523674970153, −5.93769825408826457326478754902, −5.14328431185390212290437161587, −2.80493016914380099230704351080, −1.33597101086691517606873263603, −0.54266490900032419522902891774,
2.14003214155333827571977240319, 3.18412433837487111591613276287, 5.56644789099865959743172579568, 6.99165785885809850013251625832, 7.56238595170583974932511024187, 9.599944802588823060972336104324, 10.53415825192954758111396859636, 12.07499029074647406813512402041, 13.59576071158664394141597676731, 14.87525483964641254822054223800
