| L(s) = 1 | + (0.142 + 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (0.654 − 0.755i)5-s + (0.959 + 0.281i)6-s + (2.49 − 1.60i)7-s + (−0.415 − 0.909i)8-s + (−0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (0.638 − 4.44i)11-s + (−0.142 + 0.989i)12-s + (0.235 + 0.151i)13-s + (1.94 + 2.24i)14-s + (−0.415 − 0.909i)15-s + (0.841 − 0.540i)16-s + (−6.26 − 1.84i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (0.239 − 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.292 − 0.337i)5-s + (0.391 + 0.115i)6-s + (0.944 − 0.607i)7-s + (−0.146 − 0.321i)8-s + (−0.218 − 0.251i)9-s + (0.266 + 0.170i)10-s + (0.192 − 1.33i)11-s + (−0.0410 + 0.285i)12-s + (0.0652 + 0.0419i)13-s + (0.519 + 0.600i)14-s + (−0.107 − 0.234i)15-s + (0.210 − 0.135i)16-s + (−1.52 − 0.446i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57477 - 0.641348i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57477 - 0.641348i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-2.68 + 3.97i)T \) |
| good | 7 | \( 1 + (-2.49 + 1.60i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.638 + 4.44i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.235 - 0.151i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (6.26 + 1.84i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (7.29 - 2.14i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-9.87 - 2.89i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.779 + 1.70i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-4.58 - 5.28i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.13 + 2.46i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.39 + 3.06i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + (2.13 - 1.36i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (1.55 + 0.999i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (0.890 + 1.95i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.517 - 3.59i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.27 + 8.84i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (3.71 - 1.09i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-9.73 - 6.25i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (0.201 + 0.232i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (1.94 - 4.26i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (5.46 - 6.30i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.53881282880510352160112985090, −8.949873420649293837286965088318, −8.609476062862010939967387559897, −7.84988643388413464606897508242, −6.68378868000096836359947351210, −6.17137162553585274392403713965, −4.86425030386837460917138906640, −4.10558181541513777638001001788, −2.48167627914118804771531475232, −0.880541725277278028287363722474,
1.92463667108814265507313539472, 2.60457920238053450528244048689, 4.32126968585386232544258452275, 4.63821490302584839036181332495, 5.96054839049155050865109452893, 7.05892160594253230901606046673, 8.343331291827988566919188756905, 8.962274068569990173999763177438, 9.779498467399255057569877399093, 10.74900064151707319123595211559
