| L(s) = 1 | + (71.0 − 123. i)2-s + (6.28e3 + 1.08e4i)4-s + (1.34e5 + 2.33e5i)5-s + (−2.63e5 + 4.56e5i)7-s + 6.44e6·8-s + 3.83e7·10-s + (3.21e6 − 5.57e6i)11-s + (1.01e8 + 1.76e8i)13-s + (3.74e7 + 6.48e7i)14-s + (2.51e8 − 4.36e8i)16-s − 2.81e9·17-s − 8.62e8·19-s + (−1.69e9 + 2.93e9i)20-s + (−4.57e8 − 7.92e8i)22-s + (3.34e9 + 5.79e9i)23-s + ⋯ |
| L(s) = 1 | + (0.392 − 0.679i)2-s + (0.191 + 0.332i)4-s + (0.771 + 1.33i)5-s + (−0.121 + 0.209i)7-s + 1.08·8-s + 1.21·10-s + (0.0497 − 0.0862i)11-s + (0.450 + 0.780i)13-s + (0.0949 + 0.164i)14-s + (0.234 − 0.406i)16-s − 1.66·17-s − 0.221·19-s + (−0.296 + 0.513i)20-s + (−0.0390 − 0.0677i)22-s + (0.205 + 0.355i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(2.908353503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.908353503\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 + (-71.0 + 123. i)T + (-1.63e4 - 2.83e4i)T^{2} \) |
| 5 | \( 1 + (-1.34e5 - 2.33e5i)T + (-1.52e10 + 2.64e10i)T^{2} \) |
| 7 | \( 1 + (2.63e5 - 4.56e5i)T + (-2.37e12 - 4.11e12i)T^{2} \) |
| 11 | \( 1 + (-3.21e6 + 5.57e6i)T + (-2.08e15 - 3.61e15i)T^{2} \) |
| 13 | \( 1 + (-1.01e8 - 1.76e8i)T + (-2.55e16 + 4.43e16i)T^{2} \) |
| 17 | \( 1 + 2.81e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 8.62e8T + 1.51e19T^{2} \) |
| 23 | \( 1 + (-3.34e9 - 5.79e9i)T + (-1.33e20 + 2.30e20i)T^{2} \) |
| 29 | \( 1 + (5.56e10 - 9.64e10i)T + (-4.31e21 - 7.47e21i)T^{2} \) |
| 31 | \( 1 + (1.17e11 + 2.03e11i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 - 1.11e12T + 3.33e23T^{2} \) |
| 41 | \( 1 + (3.42e11 + 5.92e11i)T + (-7.77e23 + 1.34e24i)T^{2} \) |
| 43 | \( 1 + (-7.57e11 + 1.31e12i)T + (-1.58e24 - 2.75e24i)T^{2} \) |
| 47 | \( 1 + (2.52e12 - 4.37e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 - 6.98e11T + 7.31e25T^{2} \) |
| 59 | \( 1 + (-1.26e13 - 2.18e13i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (1.34e13 - 2.32e13i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (3.16e13 + 5.48e13i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 - 3.25e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 7.84e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + (3.09e13 - 5.35e13i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + (-1.49e14 + 2.58e14i)T + (-3.05e28 - 5.29e28i)T^{2} \) |
| 89 | \( 1 + 4.09e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + (-5.02e13 + 8.69e13i)T + (-3.16e29 - 5.48e29i)T^{2} \) |
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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
−13.86288433176735965106612924536, −13.00029652123553623414202258087, −11.35424931026906474936014695858, −10.78555130027271627060439552468, −9.233264983224542330745707224902, −7.27094284348644249278589468623, −6.18190128731265070710442513502, −4.11318073749337559399484443099, −2.74884848618905091167695173045, −1.88232227314120350077103342782,
0.69560535853722404380533856868, 1.96419420300059622879969570400, 4.42931396085415363582107464809, 5.47908246116223361422563535367, 6.59780313564913549415272004974, 8.313598474249446098965027531493, 9.628239443436628849461594494849, 11.00733715906178150719742171453, 12.94290203985861755202263066730, 13.47746952574477068247375800607
