| L(s) = 1 | + (0.578 − 2.71i)2-s + (1.68 − 0.381i)3-s + (−5.23 − 2.33i)4-s + (2.07 + 0.823i)5-s + (−0.0615 − 4.81i)6-s + (−1.90 − 1.10i)7-s + (−6.09 + 8.38i)8-s + (2.70 − 1.28i)9-s + (3.44 − 5.17i)10-s + (−1.05 − 0.223i)11-s + (−9.73 − 1.93i)12-s + (0.368 + 1.73i)13-s + (−4.09 + 4.55i)14-s + (3.82 + 0.598i)15-s + (11.6 + 12.9i)16-s + (1.03 − 1.41i)17-s + ⋯ |
| L(s) = 1 | + (0.408 − 1.92i)2-s + (0.975 − 0.220i)3-s + (−2.61 − 1.16i)4-s + (0.929 + 0.368i)5-s + (−0.0251 − 1.96i)6-s + (−0.721 − 0.416i)7-s + (−2.15 + 2.96i)8-s + (0.902 − 0.429i)9-s + (1.08 − 1.63i)10-s + (−0.317 − 0.0674i)11-s + (−2.80 − 0.559i)12-s + (0.102 + 0.480i)13-s + (−1.09 + 1.21i)14-s + (0.988 + 0.154i)15-s + (2.90 + 3.22i)16-s + (0.249 − 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.368026 - 1.74387i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.368026 - 1.74387i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.68 + 0.381i)T \) |
| 5 | \( 1 + (-2.07 - 0.823i)T \) |
| good | 2 | \( 1 + (-0.578 + 2.71i)T + (-1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (1.90 + 1.10i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.05 + 0.223i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.368 - 1.73i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-1.03 + 1.41i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.98 - 1.44i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.76 + 1.58i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.934 - 8.88i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.129 + 1.22i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-6.36 - 2.06i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (4.99 - 1.06i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (2.57 + 1.48i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (10.6 - 1.11i)T + (45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (7.00 + 9.63i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.357 - 0.0760i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-3.80 - 0.807i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-4.40 - 0.462i)T + (65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (5.58 - 4.05i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.581 - 0.188i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.304 + 2.89i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-1.72 - 3.86i)T + (-55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-0.673 - 2.07i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (3.31 - 0.348i)T + (94.8 - 20.1i)T^{2} \) |
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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
−11.93177814680368652967654331335, −10.74743102443686164010667007375, −9.861727901490882774300228056870, −9.516625361722564592869442700599, −8.365979190058118959624110750090, −6.58439677137257108904967049908, −5.05157343017912116501704381063, −3.60643919764502784311121820029, −2.81569143077014163437484016406, −1.56352771987340491482389825056,
3.10623828998826049958330108790, 4.52469485547731178513283210654, 5.63081459281353474526086091775, 6.48023893897493230955851956409, 7.73245907765769308042460838620, 8.469941451660919849605990068939, 9.488778284538374896758521695969, 9.895647306740798842756132691691, 12.48237624507128025954554827595, 13.20634831556699461083922697594
