| L(s) = 1 | + (−1.17 − 0.521i)2-s + (−0.352 + 3.35i)3-s + (−0.240 − 0.266i)4-s − 1.20·5-s + (2.16 − 3.74i)6-s + (1.07 + 1.19i)7-s + (0.933 + 2.87i)8-s + (−8.20 − 1.74i)9-s + (1.41 + 0.629i)10-s + (−1.85 + 0.394i)11-s + (0.980 − 0.712i)12-s + (−3.57 − 0.465i)13-s + (−0.634 − 1.95i)14-s + (0.426 − 4.05i)15-s + (0.329 − 3.13i)16-s + (5.35 + 1.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.827 − 0.368i)2-s + (−0.203 + 1.93i)3-s + (−0.120 − 0.133i)4-s − 0.540·5-s + (0.882 − 1.52i)6-s + (0.405 + 0.450i)7-s + (0.330 + 1.01i)8-s + (−2.73 − 0.581i)9-s + (0.447 + 0.199i)10-s + (−0.559 + 0.118i)11-s + (0.282 − 0.205i)12-s + (−0.991 − 0.129i)13-s + (−0.169 − 0.521i)14-s + (0.110 − 1.04i)15-s + (0.0824 − 0.784i)16-s + (1.29 + 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0402566 - 0.0795692i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0402566 - 0.0795692i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (3.57 + 0.465i)T \) |
| 31 | \( 1 + (3.49 + 4.33i)T \) |
| good | 2 | \( 1 + (1.17 + 0.521i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (0.352 - 3.35i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + 1.20T + 5T^{2} \) |
| 7 | \( 1 + (-1.07 - 1.19i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (1.85 - 0.394i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (-5.35 - 1.13i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.302 + 2.87i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.225 - 0.250i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-3.20 - 1.42i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (1.25 + 2.17i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.14 + 1.84i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.599 + 5.70i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (5.94 + 4.32i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.36 - 10.3i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (5.64 - 2.51i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (5.14 - 8.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.481 - 0.834i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.59 - 0.339i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (5.25 - 16.1i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.761 - 2.34i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.98 + 2.16i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-8.27 + 1.75i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (3.93 + 4.37i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−11.53262363717419212031331350872, −10.55468128844245154329533136537, −10.12793524970414460961657001039, −9.338606278701943623820200726172, −8.572387625538331538688386848795, −7.71642915840225458693832790270, −5.55030365661295486781697555951, −5.10545550880494119549238211703, −4.01990755795000952413476571784, −2.65280581809620935413180454837,
0.080427264959780238971615022607, 1.49320601505773775298622338473, 3.20104904601766181125945787381, 5.03757571890550025267934450930, 6.37651001758222073564826277368, 7.29605997659104356382863367339, 7.935502191697604938745831384049, 8.139813386713961593111600276652, 9.568681926160276599832680609644, 10.71970128462631126761245647095
