L(s) = 1 | + (0.5 − 0.866i)2-s + (0.500 + 0.866i)4-s + (1.5 − 2.59i)5-s + (−2 − 1.73i)7-s + 3·8-s + (−1.5 − 2.59i)10-s + (3 + 5.19i)11-s + 13-s + (−2.5 + 0.866i)14-s + (0.500 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (2 − 3.46i)19-s + 3·20-s + 6·22-s + (2 − 3.46i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + (0.670 − 1.16i)5-s + (−0.755 − 0.654i)7-s + 1.06·8-s + (−0.474 − 0.821i)10-s + (0.904 + 1.56i)11-s + 0.277·13-s + (−0.668 + 0.231i)14-s + (0.125 − 0.216i)16-s + (−0.121 − 0.210i)17-s + (0.458 − 0.794i)19-s + 0.670·20-s + 1.27·22-s + (0.417 − 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.479321056\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.479321056\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7T + 29T^{2} \) |
| 31 | \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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
−9.767731668144289755730142869259, −9.141596268552397601724739856327, −8.165829775413167881415288830911, −6.99493873357664471222860795373, −6.59957024545222033655778502661, −5.00556484727790607526042897580, −4.47933589507849181220015900441, −3.48383452675271022971091152555, −2.22103824551955356977735315487, −1.15807201173852546747242214322,
1.49919855892003166702434593954, 2.90577556172763788980181151164, 3.65890322653304908679200282535, 5.33767484341052500650891695341, 5.98594335069302067964830398988, 6.47281252993319693230881983875, 7.11728869835905466669126672404, 8.393056621694735950159986182568, 9.257128849302038995483283601708, 10.12237919082952657319622433503