| L(s) = 1 | + (−7.65 − 13.2i)2-s + (1.62e4 − 2.81e4i)4-s + (−6.21e4 + 1.07e5i)5-s + (2.14e5 + 3.70e5i)7-s − 9.99e5·8-s + 1.90e6·10-s + (−6.06e6 − 1.04e7i)11-s + (8.18e7 − 1.41e8i)13-s + (3.27e6 − 5.67e6i)14-s + (−5.25e8 − 9.09e8i)16-s + 1.40e9·17-s − 2.81e9·19-s + (2.02e9 + 3.50e9i)20-s + (−9.28e7 + 1.60e8i)22-s + (−3.78e9 + 6.56e9i)23-s + ⋯ |
| L(s) = 1 | + (−0.0422 − 0.0732i)2-s + (0.496 − 0.859i)4-s + (−0.355 + 0.616i)5-s + (0.0982 + 0.170i)7-s − 0.168·8-s + 0.0601·10-s + (−0.0937 − 0.162i)11-s + (0.361 − 0.626i)13-s + (0.00831 − 0.0143i)14-s + (−0.489 − 0.847i)16-s + 0.830·17-s − 0.722·19-s + (0.353 + 0.611i)20-s + (−0.00793 + 0.0137i)22-s + (−0.231 + 0.401i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.124720074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.124720074\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 + (7.65 + 13.2i)T + (-1.63e4 + 2.83e4i)T^{2} \) |
| 5 | \( 1 + (6.21e4 - 1.07e5i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 7 | \( 1 + (-2.14e5 - 3.70e5i)T + (-2.37e12 + 4.11e12i)T^{2} \) |
| 11 | \( 1 + (6.06e6 + 1.04e7i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 + (-8.18e7 + 1.41e8i)T + (-2.55e16 - 4.43e16i)T^{2} \) |
| 17 | \( 1 - 1.40e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 2.81e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + (3.78e9 - 6.56e9i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 + (7.16e10 + 1.24e11i)T + (-4.31e21 + 7.47e21i)T^{2} \) |
| 31 | \( 1 + (-7.96e10 + 1.38e11i)T + (-1.17e22 - 2.03e22i)T^{2} \) |
| 37 | \( 1 + 5.75e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + (7.75e11 - 1.34e12i)T + (-7.77e23 - 1.34e24i)T^{2} \) |
| 43 | \( 1 + (1.35e12 + 2.34e12i)T + (-1.58e24 + 2.75e24i)T^{2} \) |
| 47 | \( 1 + (3.19e12 + 5.53e12i)T + (-6.03e24 + 1.04e25i)T^{2} \) |
| 53 | \( 1 + 1.65e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + (3.98e12 - 6.90e12i)T + (-1.82e26 - 3.16e26i)T^{2} \) |
| 61 | \( 1 + (1.05e13 + 1.83e13i)T + (-3.01e26 + 5.21e26i)T^{2} \) |
| 67 | \( 1 + (-2.63e13 + 4.56e13i)T + (-1.23e27 - 2.13e27i)T^{2} \) |
| 71 | \( 1 + 4.26e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 4.22e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + (-1.49e14 - 2.59e14i)T + (-1.45e28 + 2.52e28i)T^{2} \) |
| 83 | \( 1 + (-3.64e13 - 6.31e13i)T + (-3.05e28 + 5.29e28i)T^{2} \) |
| 89 | \( 1 - 5.72e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + (6.51e13 + 1.12e14i)T + (-3.16e29 + 5.48e29i)T^{2} \) |
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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
−13.54282446435184723366936414491, −11.84221364015549812563068888591, −10.84466804082716278994367350318, −9.797153642952588022320072620560, −8.009222931393097975499660033956, −6.57664928025347638049269845287, −5.37071657482185437792013910013, −3.38646070784492244076769366049, −1.89465991859422242324917712552, −0.31348837638129087462610758571,
1.56146774161980770466375538654, 3.28091255108184574661529089482, 4.63111995419160642022997281715, 6.55232159592119892918949482033, 7.87178029840093726176934369785, 8.899950053822028462641588187228, 10.73451558825436914447973399599, 12.04191892640347127981478759420, 12.82004349066191167759563277295, 14.37598118629474823067742254820
