| L(s) = 1 | + (−2.24 − 0.477i)2-s + (1.49 + 0.870i)3-s + (3.00 + 1.33i)4-s + (−2.07 − 0.827i)5-s + (−2.95 − 2.67i)6-s + (−1.62 + 2.80i)7-s + (−2.39 − 1.73i)8-s + (1.48 + 2.60i)9-s + (4.27 + 2.85i)10-s + (−1.90 − 0.405i)11-s + (3.33 + 4.61i)12-s + (−3.48 + 0.740i)13-s + (4.98 − 5.53i)14-s + (−2.39 − 3.04i)15-s + (0.148 + 0.164i)16-s + (0.726 + 0.527i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 − 0.337i)2-s + (0.864 + 0.502i)3-s + (1.50 + 0.668i)4-s + (−0.929 − 0.369i)5-s + (−1.20 − 1.09i)6-s + (−0.612 + 1.06i)7-s + (−0.845 − 0.613i)8-s + (0.494 + 0.869i)9-s + (1.35 + 0.902i)10-s + (−0.575 − 0.122i)11-s + (0.961 + 1.33i)12-s + (−0.966 + 0.205i)13-s + (1.33 − 1.47i)14-s + (−0.617 − 0.786i)15-s + (0.0370 + 0.0411i)16-s + (0.176 + 0.128i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.196058 + 0.325183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.196058 + 0.325183i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.49 - 0.870i)T \) |
| 5 | \( 1 + (2.07 + 0.827i)T \) |
| good | 2 | \( 1 + (2.24 + 0.477i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (1.62 - 2.80i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.90 + 0.405i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (3.48 - 0.740i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.726 - 0.527i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.733 - 0.532i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (5.68 - 6.31i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.780 - 7.42i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 7.70i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (1.33 - 4.10i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (4.43 - 0.943i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (0.885 - 1.53i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.808 + 7.69i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-5.94 + 4.31i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-13.5 + 2.87i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-6.17 - 1.31i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-1.26 + 12.0i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (8.02 - 5.83i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.246 - 0.759i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.0254 - 0.242i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-4.22 + 1.88i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.11 - 6.50i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.495 - 4.71i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.16328107637937297317860969241, −11.46518742571345238276598471505, −10.14901730706534716356117845711, −9.597281450958221586157698890316, −8.690435523138988228834028842760, −8.050103526135836527659116530089, −7.18840966839949204056474886386, −5.17646480234428367347796546549, −3.44871082715197262175548498498, −2.20692622128677742406819784154,
0.44967044999245410217582355835, 2.58317960592341204329215513174, 4.09857137162714868859324286947, 6.61036181405363466764514557436, 7.30378920647229239616879426955, 7.895711527400149778122948746591, 8.745991943104045191189210648171, 10.13170406775132600824571991514, 10.26306625825520263410398503047, 11.79533586018619534936776294927
