| L(s) = 1 | + (0.672 − 0.323i)2-s + (0.453 − 1.67i)3-s + (−0.899 + 1.12i)4-s + (−3.38 + 1.04i)5-s + (−0.236 − 1.27i)6-s + (1.93 + 1.80i)7-s + (−0.571 + 2.50i)8-s + (−2.58 − 1.51i)9-s + (−1.93 + 1.79i)10-s + (−0.0201 + 0.268i)11-s + (1.47 + 2.01i)12-s + (−0.418 + 5.59i)13-s + (1.88 + 0.591i)14-s + (0.212 + 6.13i)15-s + (−0.215 − 0.945i)16-s + (1.31 − 3.35i)17-s + ⋯ |
| L(s) = 1 | + (0.475 − 0.228i)2-s + (0.261 − 0.965i)3-s + (−0.449 + 0.564i)4-s + (−1.51 + 0.467i)5-s + (−0.0966 − 0.518i)6-s + (0.729 + 0.683i)7-s + (−0.202 + 0.885i)8-s + (−0.863 − 0.504i)9-s + (−0.612 + 0.568i)10-s + (−0.00607 + 0.0810i)11-s + (0.426 + 0.581i)12-s + (−0.116 + 1.55i)13-s + (0.503 + 0.157i)14-s + (0.0547 + 1.58i)15-s + (−0.0539 − 0.236i)16-s + (0.319 − 0.813i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.721627 + 0.633783i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.721627 + 0.633783i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.453 + 1.67i)T \) |
| 7 | \( 1 + (-1.93 - 1.80i)T \) |
| good | 2 | \( 1 + (-0.672 + 0.323i)T + (1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (3.38 - 1.04i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.0201 - 0.268i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (0.418 - 5.59i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-1.31 + 3.35i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (3.46 - 6.00i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.26 - 0.492i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (2.71 - 6.91i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 + (-6.61 + 0.997i)T + (35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (8.53 + 7.91i)T + (3.06 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-4.24 + 3.93i)T + (3.21 - 42.8i)T^{2} \) |
| 47 | \( 1 + (-2.55 + 1.23i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (-10.3 - 1.55i)T + (50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (1.07 + 4.72i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (0.376 + 0.471i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 7.93T + 67T^{2} \) |
| 71 | \( 1 + (7.35 - 9.21i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.188 - 2.51i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + 2.08T + 79T^{2} \) |
| 83 | \( 1 + (-0.364 - 4.85i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (0.493 - 0.336i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (-6.56 - 11.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.70702806360469039828418631566, −11.00039385567211327537451475741, −9.015050328433042053077955754217, −8.556039200609127951639018214963, −7.55912537310976194108163690073, −7.07391221074789548083287367896, −5.50409119535500836571493396712, −4.25471163570584427390756304510, −3.38759878853966941743693209006, −2.09900850276629566557500492609,
0.51297681286287455808738530755, 3.30796618968322288785057767252, 4.31125038553735921658165099458, 4.73307642463650130803687607168, 5.81218641260613068016138264918, 7.46163380765294271197514395551, 8.218320092245363169004237734214, 8.979387430568400385959300867760, 10.19960904724946373305595015320, 10.85189847159132804075948159351
