| L(s) = 1 | + (−0.0741 + 0.847i)2-s + (−0.344 − 0.737i)3-s + (1.25 + 0.221i)4-s + (0.650 − 0.236i)6-s + (3.83 − 1.02i)7-s + (−0.721 + 2.69i)8-s + (1.50 − 1.79i)9-s + (−2.45 − 4.25i)11-s + (−0.269 − 1.00i)12-s + (−1.10 − 0.515i)13-s + (0.586 + 3.32i)14-s + (0.173 + 0.0630i)16-s + (1.40 + 0.122i)17-s + (1.40 + 1.40i)18-s + (−4.22 + 1.07i)19-s + ⋯ |
| L(s) = 1 | + (−0.0524 + 0.598i)2-s + (−0.198 − 0.426i)3-s + (0.628 + 0.110i)4-s + (0.265 − 0.0966i)6-s + (1.45 − 0.388i)7-s + (−0.254 + 0.951i)8-s + (0.500 − 0.596i)9-s + (−0.740 − 1.28i)11-s + (−0.0776 − 0.289i)12-s + (−0.306 − 0.143i)13-s + (0.156 + 0.889i)14-s + (0.0433 + 0.0157i)16-s + (0.340 + 0.0297i)17-s + (0.331 + 0.331i)18-s + (−0.968 + 0.247i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73375 + 0.0391624i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73375 + 0.0391624i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (4.22 - 1.07i)T \) |
| good | 2 | \( 1 + (0.0741 - 0.847i)T + (-1.96 - 0.347i)T^{2} \) |
| 3 | \( 1 + (0.344 + 0.737i)T + (-1.92 + 2.29i)T^{2} \) |
| 7 | \( 1 + (-3.83 + 1.02i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (2.45 + 4.25i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.10 + 0.515i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-1.40 - 0.122i)T + (16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (-0.867 - 1.23i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-1.63 - 1.37i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-8.01 - 4.62i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.74 + 6.74i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.783 + 2.15i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.09 - 0.769i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-0.434 - 4.96i)T + (-46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (7.59 - 5.31i)T + (18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (9.46 - 7.94i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.09 - 6.20i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (8.76 - 0.766i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (2.67 - 0.471i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.139 + 0.0649i)T + (46.9 - 55.9i)T^{2} \) |
| 79 | \( 1 + (-1.57 - 0.574i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.200 + 0.747i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (3.86 - 1.40i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.860 - 9.84i)T + (-95.5 - 16.8i)T^{2} \) |
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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
−10.98808052739100161114339596083, −10.47043370229784826271985632250, −8.861176054906557484344240660478, −7.950753748983195662212239612119, −7.53296951625122951846298390253, −6.38454948909942123034868430097, −5.62641169172894831446913391972, −4.43422640888372389396520427542, −2.81874661822914918328961984020, −1.31019861671314860493432665133,
1.73904358236532530862936631012, 2.54699514291410717310535016407, 4.46388349117402170034872981491, 4.92242022122763759646725472020, 6.33769731561970533159699880212, 7.53410325490540544551882693955, 8.139506956042776712046883661974, 9.646715174938986948732264096183, 10.24717735812669560190786540593, 11.00059245634711374471758165495
