| L(s) = 1 | + (0.900 − 0.433i)2-s + (0.0351 + 1.73i)3-s + (0.623 − 0.781i)4-s + (−2.94 + 0.908i)5-s + (0.783 + 1.54i)6-s + (−1.59 − 2.10i)7-s + (0.222 − 0.974i)8-s + (−2.99 + 0.121i)9-s + (−2.25 + 2.09i)10-s + (−0.310 + 4.14i)11-s + (1.37 + 1.05i)12-s + (0.420 − 5.60i)13-s + (−2.35 − 1.20i)14-s + (−1.67 − 5.06i)15-s + (−0.222 − 0.974i)16-s + (1.61 − 4.11i)17-s + ⋯ |
| L(s) = 1 | + (0.637 − 0.306i)2-s + (0.0203 + 0.999i)3-s + (0.311 − 0.390i)4-s + (−1.31 + 0.406i)5-s + (0.319 + 0.630i)6-s + (−0.603 − 0.797i)7-s + (0.0786 − 0.344i)8-s + (−0.999 + 0.0406i)9-s + (−0.714 + 0.662i)10-s + (−0.0937 + 1.25i)11-s + (0.397 + 0.303i)12-s + (0.116 − 1.55i)13-s + (−0.629 − 0.322i)14-s + (−0.432 − 1.30i)15-s + (−0.0556 − 0.243i)16-s + (0.392 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.454629 - 0.648295i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.454629 - 0.648295i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.900 + 0.433i)T \) |
| 3 | \( 1 + (-0.0351 - 1.73i)T \) |
| 7 | \( 1 + (1.59 + 2.10i)T \) |
| good | 5 | \( 1 + (2.94 - 0.908i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.310 - 4.14i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (-0.420 + 5.60i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-1.61 + 4.11i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (-3.38 + 5.85i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.76 + 0.567i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-1.68 + 4.30i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + 0.590T + 31T^{2} \) |
| 37 | \( 1 + (6.18 - 0.931i)T + (35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (-3.83 - 3.56i)T + (3.06 + 40.8i)T^{2} \) |
| 43 | \( 1 + (4.97 - 4.61i)T + (3.21 - 42.8i)T^{2} \) |
| 47 | \( 1 + (8.44 - 4.06i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (5.81 + 0.877i)T + (50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (-0.383 - 1.67i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (6.86 + 8.60i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 8.21T + 67T^{2} \) |
| 71 | \( 1 + (1.20 - 1.50i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (0.216 + 2.88i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + (0.833 + 11.1i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-13.4 + 9.16i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 + (-1.51 - 2.62i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.03373797273207865275785258232, −9.442674369413282502173086319679, −7.919123472327882294075127774956, −7.44373300389106601386927876204, −6.36168731136741965428300370082, −4.99305487816876271149636040259, −4.48412637064936019462533973404, −3.37154436978758903349694051776, −2.96021728720367275166072658274, −0.30059891411875993806238446077,
1.69685412751850316618338280329, 3.29483203970713258091785826064, 3.84629920025557918416296475278, 5.35320774555883710938729112179, 6.10802666766569767509096460920, 6.86201144938841838495212782272, 7.930085905322801839756948476563, 8.379823066690388150069082251138, 9.153374191900153318256533718457, 10.71008476980494071210586616220
