| L(s) = 1 | + (0.0510 − 0.354i)2-s + (0.415 + 0.909i)3-s + (1.79 + 0.527i)4-s + (0.654 + 0.755i)5-s + (0.343 − 0.100i)6-s + (2.04 + 1.31i)7-s + (0.576 − 1.26i)8-s + (−0.654 + 0.755i)9-s + (0.301 − 0.193i)10-s + (−0.621 − 4.32i)11-s + (0.266 + 1.85i)12-s + (−4.84 + 3.11i)13-s + (0.571 − 0.659i)14-s + (−0.415 + 0.909i)15-s + (2.73 + 1.75i)16-s + (−0.233 + 0.0686i)17-s + ⋯ |
| L(s) = 1 | + (0.0360 − 0.250i)2-s + (0.239 + 0.525i)3-s + (0.897 + 0.263i)4-s + (0.292 + 0.337i)5-s + (0.140 − 0.0412i)6-s + (0.774 + 0.497i)7-s + (0.203 − 0.446i)8-s + (−0.218 + 0.251i)9-s + (0.0953 − 0.0612i)10-s + (−0.187 − 1.30i)11-s + (0.0768 + 0.534i)12-s + (−1.34 + 0.862i)13-s + (0.152 − 0.176i)14-s + (−0.107 + 0.234i)15-s + (0.682 + 0.438i)16-s + (−0.0566 + 0.0166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.80989 + 0.494398i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80989 + 0.494398i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-4.21 + 2.28i)T \) |
| good | 2 | \( 1 + (-0.0510 + 0.354i)T + (-1.91 - 0.563i)T^{2} \) |
| 7 | \( 1 + (-2.04 - 1.31i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.621 + 4.32i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (4.84 - 3.11i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.233 - 0.0686i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (2.95 + 0.866i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (2.46 - 0.725i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.55 + 3.39i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-3.05 + 3.52i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (3.15 + 3.63i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.06 - 8.89i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 5.51T + 47T^{2} \) |
| 53 | \( 1 + (-3.62 - 2.33i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (4.37 - 2.80i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.89 + 8.53i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.238 - 1.65i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.59 + 11.1i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (6.71 + 1.97i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (5.09 - 3.27i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (8.27 - 9.54i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (5.27 + 11.5i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (9.55 + 11.0i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.27226407473101550652630785146, −10.98821834912612178738547267654, −9.843082693050175236062161918228, −8.828414988029548606981565842983, −7.87676174216215309402289475149, −6.80270238548949120552627192424, −5.71544161814443689789788234033, −4.48445497325804119282273388006, −3.02049392676055771239367543339, −2.12113706369525584186165545770,
1.57210974169405643497135044278, 2.65132037678907062340439673191, 4.65703118486862398374016551596, 5.53298292030468355548890754942, 6.94220698272063882602864556576, 7.44132307773300243348339337852, 8.318287401014100699445757044919, 9.762481275679208809134540265793, 10.44727864823879717870500121043, 11.51422051671457163329487928675
