| L(s) = 1 | + 4i·2-s − 16·4-s + (16.1 − 53.5i)5-s + 119. i·7-s − 64i·8-s + (214. + 64.6i)10-s − 263.·11-s + 851. i·13-s − 478.·14-s + 256·16-s + 1.28e3i·17-s − 2.06e3·19-s + (−258. + 856. i)20-s − 1.05e3i·22-s − 55.5i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.289 − 0.957i)5-s + 0.923i·7-s − 0.353i·8-s + (0.676 + 0.204i)10-s − 0.657·11-s + 1.39i·13-s − 0.652·14-s + 0.250·16-s + 1.08i·17-s − 1.30·19-s + (−0.144 + 0.478i)20-s − 0.464i·22-s − 0.0218i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.131211 + 0.887802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.131211 + 0.887802i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-16.1 + 53.5i)T \) |
| good | 7 | \( 1 - 119. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 263.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 851. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.28e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.06e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 55.5iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.21e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.85e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.18e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 5.59e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.64e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.43e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.74e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.69e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 6.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.27e4iT - 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.56492101969616244110897990923, −12.75891571518617041096736989949, −11.73096527327456472970579143158, −10.08181451864689316756089210732, −8.873603250973746200751224253913, −8.293252091615239156653898837897, −6.56334198889204564181939122023, −5.48608843569509357909648841030, −4.28847035841619356166657976037, −1.91551206292560839052815045964,
0.34843230763806503579648593451, 2.40996843467623513214830140479, 3.67859136491351877495478086447, 5.36709363232569695252305756114, 6.96922050669780190298297055573, 8.113147686911082629732938139271, 9.811320181861855733388042671453, 10.53911234730909633881610146738, 11.27561918413671922386018167145, 12.81661387686208103615871067035
