L(s) = 1 | − 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−2.25 − 3.91i)7-s − 8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (1.93 − 3.34i)11-s + (0.5 + 0.866i)12-s + (−3.51 + 6.09i)13-s + (2.25 + 3.91i)14-s − 0.999·15-s + 16-s + (−0.240 − 0.416i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (−0.853 − 1.47i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s + (0.582 − 1.00i)11-s + (0.144 + 0.249i)12-s + (−0.975 + 1.69i)13-s + (0.603 + 1.04i)14-s − 0.258·15-s + 0.250·16-s + (−0.0583 − 0.101i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.965169 + 0.410170i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.965169 + 0.410170i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-5.12 + 2.18i)T \) |
good | 7 | \( 1 + (2.25 + 3.91i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.93 + 3.34i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.51 - 6.09i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.240 + 0.416i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.95 - 6.84i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5.51T + 23T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 37 | \( 1 + (-1.24 - 2.14i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.08 - 7.07i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.19 - 3.79i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + (-0.827 + 1.43i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.77 + 11.7i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 5.90T + 61T^{2} \) |
| 67 | \( 1 + (1.24 - 2.14i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.51 - 2.63i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.19 - 3.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.603 + 1.04i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 2.96T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.867541744743541015239229162270, −9.672656061267516850946142168260, −8.584286315165619357069383279542, −7.66138221535771215277720319593, −6.84731924892785659123597793362, −6.28669904656011324066190425969, −4.61696048569863638550608068796, −3.72151736743783107257045965508, −2.88299940335753496880303075994, −1.06979740902724276211736670392,
0.75966601704925162593160717689, 2.52631870974947639674155129157, 2.97816548262144184881540877917, 4.86245588915534514469288005656, 5.70676125520593317048438353516, 6.85841391313145261818285440066, 7.38802544811058539426543702022, 8.522287413076362353664015838453, 9.040581798544188018434915490333, 9.708857846243883987150495308754