| L(s) = 1 | + (1.02 − 0.972i)2-s + (−0.973 + 1.43i)3-s + (0.106 − 1.99i)4-s + (−1.75 + 0.153i)5-s + (0.394 + 2.41i)6-s + (0.584 − 1.60i)7-s + (−1.83 − 2.15i)8-s + (−1.10 − 2.78i)9-s + (−1.65 + 1.86i)10-s + (0.287 − 3.28i)11-s + (2.75 + 2.09i)12-s + (−0.848 + 0.593i)13-s + (−0.962 − 2.21i)14-s + (1.48 − 2.66i)15-s + (−3.97 − 0.425i)16-s + (2.08 − 3.60i)17-s + ⋯ |
| L(s) = 1 | + (0.725 − 0.687i)2-s + (−0.562 + 0.827i)3-s + (0.0533 − 0.998i)4-s + (−0.784 + 0.0686i)5-s + (0.161 + 0.986i)6-s + (0.220 − 0.607i)7-s + (−0.648 − 0.761i)8-s + (−0.368 − 0.929i)9-s + (−0.522 + 0.589i)10-s + (0.0867 − 0.991i)11-s + (0.795 + 0.605i)12-s + (−0.235 + 0.164i)13-s + (−0.257 − 0.592i)14-s + (0.384 − 0.687i)15-s + (−0.994 − 0.106i)16-s + (0.505 − 0.874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.556475 - 1.02250i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.556475 - 1.02250i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.02 + 0.972i)T \) |
| 3 | \( 1 + (0.973 - 1.43i)T \) |
| good | 5 | \( 1 + (1.75 - 0.153i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-0.584 + 1.60i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.287 + 3.28i)T + (-10.8 - 1.91i)T^{2} \) |
| 13 | \( 1 + (0.848 - 0.593i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-2.08 + 3.60i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0883 - 0.329i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (2.29 + 6.30i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.05 + 1.51i)T + (-9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (-1.37 + 0.500i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.02 - 11.2i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.89 + 0.333i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.863 + 9.86i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-0.0746 - 0.0271i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (4.23 - 4.23i)T - 53iT^{2} \) |
| 59 | \( 1 + (7.47 - 0.653i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (-6.57 - 14.0i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-4.27 + 2.99i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-5.29 - 3.05i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-9.72 + 5.61i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.776 + 4.40i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.70 + 2.43i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-12.0 + 6.94i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.75 - 2.31i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.90186514028679472819239677411, −10.32792273896411225944644172908, −9.386362374715364651828734080765, −8.205806498910571756688192674388, −6.81638380719275759817730550981, −5.81005462113788136288335681664, −4.71540348487783210990746811109, −3.98960666664092638354040891917, −2.99363626026222529239364849309, −0.62647137982542855286540009447,
2.11798358731189161982864244167, 3.74018348772146024467422357787, 4.93942619385850665255575956579, 5.77930972138100081386700384507, 6.76803759527789648153944130762, 7.77292440727370677936956397391, 8.087648664759275523595946540589, 9.509293878179005859911026119347, 11.03251014303092851310115779699, 11.79044930216346019349501303041
