| L(s) = 1 | + (−146. + 253. i)2-s + (−2.64e4 − 4.58e4i)4-s + (2.82e4 + 4.89e4i)5-s + (−1.25e5 + 2.16e5i)7-s + 5.90e6·8-s − 1.65e7·10-s + (−4.17e7 + 7.22e7i)11-s + (1.70e8 + 2.95e8i)13-s + (−3.66e7 − 6.34e7i)14-s + (3.01e6 − 5.21e6i)16-s − 3.82e8·17-s − 7.33e9·19-s + (1.49e9 − 2.59e9i)20-s + (−1.22e10 − 2.11e10i)22-s + (−3.07e9 − 5.32e9i)23-s + ⋯ |
| L(s) = 1 | + (−0.808 + 1.40i)2-s + (−0.807 − 1.39i)4-s + (0.161 + 0.280i)5-s + (−0.0574 + 0.0995i)7-s + 0.995·8-s − 0.523·10-s + (−0.645 + 1.11i)11-s + (0.753 + 1.30i)13-s + (−0.0929 − 0.160i)14-s + (0.00280 − 0.00485i)16-s − 0.226·17-s − 1.88·19-s + (0.261 − 0.453i)20-s + (−1.04 − 1.80i)22-s + (−0.188 − 0.325i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.01887021423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01887021423\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 + (146. - 253. i)T + (-1.63e4 - 2.83e4i)T^{2} \) |
| 5 | \( 1 + (-2.82e4 - 4.89e4i)T + (-1.52e10 + 2.64e10i)T^{2} \) |
| 7 | \( 1 + (1.25e5 - 2.16e5i)T + (-2.37e12 - 4.11e12i)T^{2} \) |
| 11 | \( 1 + (4.17e7 - 7.22e7i)T + (-2.08e15 - 3.61e15i)T^{2} \) |
| 13 | \( 1 + (-1.70e8 - 2.95e8i)T + (-2.55e16 + 4.43e16i)T^{2} \) |
| 17 | \( 1 + 3.82e8T + 2.86e18T^{2} \) |
| 19 | \( 1 + 7.33e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + (3.07e9 + 5.32e9i)T + (-1.33e20 + 2.30e20i)T^{2} \) |
| 29 | \( 1 + (5.63e10 - 9.75e10i)T + (-4.31e21 - 7.47e21i)T^{2} \) |
| 31 | \( 1 + (7.82e10 + 1.35e11i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 - 1.00e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + (-6.24e11 - 1.08e12i)T + (-7.77e23 + 1.34e24i)T^{2} \) |
| 43 | \( 1 + (-4.26e11 + 7.38e11i)T + (-1.58e24 - 2.75e24i)T^{2} \) |
| 47 | \( 1 + (-1.31e12 + 2.27e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 + 4.33e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + (1.60e13 + 2.77e13i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (-1.45e13 + 2.51e13i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (-2.19e13 - 3.80e13i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 + 2.74e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 2.89e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + (-1.04e14 + 1.80e14i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + (-1.12e14 + 1.95e14i)T + (-3.05e28 - 5.29e28i)T^{2} \) |
| 89 | \( 1 + 7.31e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + (-2.06e14 + 3.57e14i)T + (-3.16e29 - 5.48e29i)T^{2} \) |
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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
−14.29936305449013087897857677885, −12.71112199385641242105581214618, −10.78641875054738466581965584849, −9.456071565381350312452194719071, −8.389468642294388296089772603536, −7.03451991545075999338935305958, −6.16382505093091752421902997559, −4.49249865963253963660319618132, −2.02274490143347457724402363794, −0.008276424614857764645799641491,
1.03748168555116949798762223320, 2.50947851778755285758300368109, 3.73858297132736309734895552713, 5.79664707330306080099436048409, 8.102897873441249982837039382565, 8.986722161357410811886976905228, 10.51050853505732024552433649394, 11.04910473891502889992623297518, 12.65033813592589761613666788875, 13.37288270350110564482483987397
