| L(s) = 1 | + (−0.355 − 1.36i)2-s + (0.540 − 0.841i)3-s + (−1.74 + 0.973i)4-s + (0.190 + 0.0867i)5-s + (−1.34 − 0.440i)6-s + (2.41 + 0.710i)7-s + (1.95 + 2.04i)8-s + (−0.415 − 0.909i)9-s + (0.0511 − 0.290i)10-s + (4.36 + 3.78i)11-s + (−0.125 + 1.99i)12-s + (1.69 + 5.78i)13-s + (0.111 − 3.56i)14-s + (0.175 − 0.112i)15-s + (2.10 − 3.40i)16-s + (0.569 + 3.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.251 − 0.967i)2-s + (0.312 − 0.485i)3-s + (−0.873 + 0.486i)4-s + (0.0849 + 0.0388i)5-s + (−0.548 − 0.179i)6-s + (0.914 + 0.268i)7-s + (0.690 + 0.722i)8-s + (−0.138 − 0.303i)9-s + (0.0161 − 0.0920i)10-s + (1.31 + 1.14i)11-s + (−0.0361 + 0.576i)12-s + (0.471 + 1.60i)13-s + (0.0298 − 0.952i)14-s + (0.0453 − 0.0291i)15-s + (0.525 − 0.850i)16-s + (0.138 + 0.960i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.40785 - 0.560399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40785 - 0.560399i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.355 + 1.36i)T \) |
| 3 | \( 1 + (-0.540 + 0.841i)T \) |
| 23 | \( 1 + (-3.93 + 2.74i)T \) |
| good | 5 | \( 1 + (-0.190 - 0.0867i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-2.41 - 0.710i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-4.36 - 3.78i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.69 - 5.78i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.569 - 3.95i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (6.83 + 0.982i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.673 + 0.0969i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.246 + 0.158i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.94 + 0.889i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-4.71 + 10.3i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (3.10 - 4.83i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 3.14T + 47T^{2} \) |
| 53 | \( 1 + (-0.724 + 2.46i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.17 - 7.41i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (6.75 + 10.5i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (2.83 - 2.45i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-7.73 - 8.92i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.62 + 11.3i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (2.27 - 0.669i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (13.0 - 5.94i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (14.0 + 9.05i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (2.56 - 5.60i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.85976301388763144893158493218, −9.746834559679570393098870833373, −8.843331590866834095358727716981, −8.447451025679435984921617485147, −7.17092234277796941726557541825, −6.25290979131618720132915626690, −4.48843315883599453354084819248, −4.04906236846845293519845923616, −2.17065120382201403706744493031, −1.61001238588799713274884310122,
1.11043523004824325187406096643, 3.35582761545422470549822946562, 4.40542057526279896460341785480, 5.44794785570299755108991950936, 6.25001489662932790237340555774, 7.49178062392198350371492869995, 8.300443263685773471545114006427, 8.878009796763036179343466155469, 9.800020867936324681846787799926, 10.81802717311732352687019868297
