L(s) = 1 | − 1.13i·2-s + 30.7·4-s + 11.6·5-s + 79.1i·7-s − 71.1i·8-s − 13.2i·10-s − 192.·11-s + 383.·13-s + 89.9·14-s + 901.·16-s + 1.91e3·17-s + 2.10e3i·19-s + 357.·20-s + 217. i·22-s + (−2.01e3 − 1.53e3i)23-s + ⋯ |
L(s) = 1 | − 0.200i·2-s + 0.959·4-s + 0.208·5-s + 0.610i·7-s − 0.393i·8-s − 0.0418i·10-s − 0.478·11-s + 0.628·13-s + 0.122·14-s + 0.880·16-s + 1.60·17-s + 1.33i·19-s + 0.199·20-s + 0.0960i·22-s + (−0.795 − 0.605i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.299i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.953 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.772846932\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.772846932\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + (2.01e3 + 1.53e3i)T \) |
good | 2 | \( 1 + 1.13iT - 32T^{2} \) |
| 5 | \( 1 - 11.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 79.1iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 192.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 383.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.91e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.10e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 3.35e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 7.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.61e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 6.21e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.57e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.57e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.52e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.51e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 5.16e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.86e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.42e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 5.87e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.16e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 4.92e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.45e4iT - 8.58e9T^{2} \) |
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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
−11.87085204675847037907200834844, −10.41944580102657326873557743413, −9.987735522061988683262518653962, −8.414068706970512523937716806994, −7.60936507098983417838558556384, −6.20760191696832807367786447931, −5.56857764870566385288049517496, −3.69160498632684222009359673244, −2.50286191337165347344798664038, −1.26899628323080561652788005622,
0.935584543890146619678084019685, 2.40801078115041272748256415675, 3.72112796603104299264593318395, 5.37128551233178440985400921165, 6.32731424379051023716722894848, 7.43983860901189462829012241430, 8.155643334091280958616324271354, 9.724885049258954779458297460015, 10.50784151844886022879831950500, 11.45429573790141830333808589798