| L(s) = 1 | − 1.41i·2-s + 3.88i·3-s − 2.00·4-s − 8.89·5-s + 5.48·6-s − 5.10·7-s + 2.82i·8-s − 6.06·9-s + 12.5i·10-s − 5.40·11-s − 7.76i·12-s + 12.5i·13-s + 7.22i·14-s − 34.5i·15-s + 4.00·16-s − 25.7·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.29i·3-s − 0.500·4-s − 1.77·5-s + 0.914·6-s − 0.729·7-s + 0.353i·8-s − 0.674·9-s + 1.25i·10-s − 0.491·11-s − 0.646i·12-s + 0.963i·13-s + 0.515i·14-s − 2.30i·15-s + 0.250·16-s − 1.51·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4063735690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4063735690\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.88iT - 9T^{2} \) |
| 5 | \( 1 + 8.89T + 25T^{2} \) |
| 7 | \( 1 + 5.10T + 49T^{2} \) |
| 11 | \( 1 + 5.40T + 121T^{2} \) |
| 13 | \( 1 - 12.5iT - 169T^{2} \) |
| 17 | \( 1 + 25.7T + 289T^{2} \) |
| 23 | \( 1 - 8.37T + 529T^{2} \) |
| 29 | \( 1 + 47.2iT - 841T^{2} \) |
| 31 | \( 1 - 22.0iT - 961T^{2} \) |
| 37 | \( 1 + 16.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 5.08iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 37.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 44.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 28.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 42.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 49.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 25.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 59.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 34.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 73.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 71.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 52.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 43.4iT - 9.40e3T^{2} \) |
| 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
−10.20003414996507997991781462952, −9.240900092717927121742503044029, −8.716159316841446120907337913915, −7.62107202020026830805177793271, −6.59117332840220936765980408802, −4.98611073908605592269158517635, −4.17429604564245643944678899758, −3.79622099463534111963161472757, −2.62881992822487436680696596201, −0.23193532455969028464761203753,
0.72322946197040830941250995296, 2.76005794005653536922017170934, 3.87004689159644145984970820383, 4.99356677035284675198624292884, 6.30641451139747598955878446389, 7.04145616153408863831738473387, 7.59094412686739955669798457596, 8.272982023070389463574017809838, 9.037099762689534937397304313396, 10.49720344841824963282138873239
