| L(s) = 1 | + 13.4i·2-s + 75.5·4-s − 968. i·5-s + 2.98e3·7-s + 4.45e3i·8-s + 1.30e4·10-s + 2.10e4i·11-s + 2.98e4·13-s + 4.00e4i·14-s − 4.04e4·16-s + 8.81e4i·17-s + 1.98e5·19-s − 7.31e4i·20-s − 2.82e5·22-s + 2.35e5i·23-s + ⋯ |
| L(s) = 1 | + 0.839i·2-s + 0.295·4-s − 1.54i·5-s + 1.24·7-s + 1.08i·8-s + 1.30·10-s + 1.43i·11-s + 1.04·13-s + 1.04i·14-s − 0.617·16-s + 1.05i·17-s + 1.52·19-s − 0.457i·20-s − 1.20·22-s + 0.841i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.294054023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.294054023\) |
| \(L(5)\) |
|
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 | \( 1 - 13.4iT - 256T^{2} \) |
| 5 | \( 1 + 968. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 2.98e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.10e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.98e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 8.81e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.98e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 2.35e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.01e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.52e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.15e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 1.01e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.29e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 7.90e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.17e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.80e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.04e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.98e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.35e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.14e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 6.12e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 3.97e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.20e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 4.10e7T + 7.83e15T^{2} \) |
| 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
−11.11943540365145370251741640330, −9.699891250385988938465598238912, −8.617407413370258168253233375375, −7.985783314981764830209700171801, −7.14819754582994588820106702361, −5.61391966123206741764424787502, −5.12457919689073764195906125227, −4.01479261986416798397778573519, −1.82343286809545189835565432957, −1.31740432271627288873029383762,
0.71385871799284664407919424150, 1.84034379221617262622985881633, 3.04408707989617941843731250588, 3.55990619852341665584668864071, 5.35715114687315713283115283440, 6.54126388010840220274866761778, 7.37871625023183843610696384425, 8.489608023479173131982653977558, 9.841688336583839495705179303348, 10.89882398556768307689560876060
