| L(s) = 1 | + (0.939 − 0.342i)2-s + (0.873 + 1.49i)3-s + (0.766 − 0.642i)4-s + (−0.605 − 3.43i)5-s + (1.33 + 1.10i)6-s + (0.766 + 0.642i)7-s + (0.500 − 0.866i)8-s + (−1.47 + 2.61i)9-s + (−1.74 − 3.02i)10-s + (0.791 − 4.48i)11-s + (1.63 + 0.584i)12-s + (5.50 + 2.00i)13-s + (0.939 + 0.342i)14-s + (4.61 − 3.90i)15-s + (0.173 − 0.984i)16-s + (0.614 + 1.06i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.504 + 0.863i)3-s + (0.383 − 0.321i)4-s + (−0.270 − 1.53i)5-s + (0.543 + 0.451i)6-s + (0.289 + 0.242i)7-s + (0.176 − 0.306i)8-s + (−0.491 + 0.870i)9-s + (−0.551 − 0.955i)10-s + (0.238 − 1.35i)11-s + (0.470 + 0.168i)12-s + (1.52 + 0.555i)13-s + (0.251 + 0.0914i)14-s + (1.19 − 1.00i)15-s + (0.0434 − 0.246i)16-s + (0.148 + 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.24484 - 0.446804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.24484 - 0.446804i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.873 - 1.49i)T \) |
| 7 | \( 1 + (-0.766 - 0.642i)T \) |
| good | 5 | \( 1 + (0.605 + 3.43i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.791 + 4.48i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-5.50 - 2.00i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.614 - 1.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.418 - 0.725i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.66 - 4.74i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.897 + 0.326i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (6.47 - 5.43i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (0.215 + 0.372i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.55 - 2.38i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.23 + 6.98i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (6.64 + 5.57i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 + (-1.92 - 10.9i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.16 - 6.84i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (13.7 + 5.01i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.0121 - 0.0211i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.20 + 2.08i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.95 - 0.712i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-15.8 + 5.75i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (3.51 - 6.08i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.34 - 13.3i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.38956460650079814782854245166, −10.58214194983212385181746212284, −9.222442976335698169114534022097, −8.703363156653164587348186115693, −7.973763883632464220447656752058, −5.98028000305893669937713877645, −5.31409197702755879781966306640, −4.12634512928819913141727461014, −3.52856419242508757189608727039, −1.54595335848860899037308531218,
2.03328152828622658864729460018, 3.20287462613151510780003030130, 4.18187369052867605971558722052, 6.03064194752000536783695823745, 6.66905823059867955070098442357, 7.53188530281137851472141530070, 8.156929523582579188892772438671, 9.643453753794098051343621678014, 10.82640198717356028755967135230, 11.43849589925672514767684228958
