L(s) = 1 | + 5.29i·5-s − 7.48i·7-s + 5.65·11-s + 4i·13-s − 21.1·17-s + 29.9·19-s + 22.6i·23-s − 3.00·25-s + 5.29i·29-s − 22.4i·31-s + 39.5·35-s − 28i·37-s + 63.4·41-s − 29.9·43-s + 67.8i·47-s + ⋯ |
L(s) = 1 | + 1.05i·5-s − 1.06i·7-s + 0.514·11-s + 0.307i·13-s − 1.24·17-s + 1.57·19-s + 0.983i·23-s − 0.120·25-s + 0.182i·29-s − 0.724i·31-s + 1.13·35-s − 0.756i·37-s + 1.54·41-s − 0.696·43-s + 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.908686082\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.908686082\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5.29iT - 25T^{2} \) |
| 7 | \( 1 + 7.48iT - 49T^{2} \) |
| 11 | \( 1 - 5.65T + 121T^{2} \) |
| 13 | \( 1 - 4iT - 169T^{2} \) |
| 17 | \( 1 + 21.1T + 289T^{2} \) |
| 19 | \( 1 - 29.9T + 361T^{2} \) |
| 23 | \( 1 - 22.6iT - 529T^{2} \) |
| 29 | \( 1 - 5.29iT - 841T^{2} \) |
| 31 | \( 1 + 22.4iT - 961T^{2} \) |
| 37 | \( 1 + 28iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 63.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 29.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 47.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 101.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 76iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 59.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 26T + 5.32e3T^{2} \) |
| 79 | \( 1 - 127. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 42.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 18T + 9.40e3T^{2} \) |
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
−9.724122004550450320114891404802, −9.089615272813618628429298752896, −7.78789026351528471565252079573, −7.16082590635146067101136517507, −6.61487777480005039099859953196, −5.54502309055337924699080904041, −4.28278064950008385723380400906, −3.57445486295418085256722010699, −2.46827739988855517484640408431, −1.02864294030720952459993727424,
0.71396711984107043857150220849, 2.01177067866957031477257909464, 3.17348423941846836989068791107, 4.49806971085420248470725414937, 5.15931633936319713759785409658, 6.02241350821158002025955337022, 6.96348067499136555681067592805, 8.098966292591681645802254191153, 8.859049871230294393473653183424, 9.191862500100757790870548624849