| L(s) = 1 | + (−1.01 − 0.698i)2-s + (−0.885 + 0.464i)3-s + (−0.173 − 0.457i)4-s + (2.41 − 2.72i)5-s + (1.21 + 0.148i)6-s + (−1.07 + 4.36i)7-s + (−0.732 + 2.97i)8-s + (0.568 − 0.822i)9-s + (−4.33 + 1.06i)10-s + (−2.95 + 2.03i)11-s + (0.366 + 0.324i)12-s + (−3.46 − 0.984i)13-s + (4.13 − 3.66i)14-s + (−0.869 + 3.52i)15-s + (2.08 − 1.84i)16-s + (0.271 + 0.0668i)17-s + ⋯ |
| L(s) = 1 | + (−0.715 − 0.493i)2-s + (−0.511 + 0.268i)3-s + (−0.0868 − 0.228i)4-s + (1.07 − 1.21i)5-s + (0.498 + 0.0604i)6-s + (−0.406 + 1.65i)7-s + (−0.258 + 1.05i)8-s + (0.189 − 0.274i)9-s + (−1.37 + 0.338i)10-s + (−0.890 + 0.614i)11-s + (0.105 + 0.0937i)12-s + (−0.961 − 0.273i)13-s + (1.10 − 0.979i)14-s + (−0.224 + 0.911i)15-s + (0.520 − 0.460i)16-s + (0.0657 + 0.0162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.390239 + 0.281499i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.390239 + 0.281499i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.885 - 0.464i)T \) |
| 13 | \( 1 + (3.46 + 0.984i)T \) |
| good | 2 | \( 1 + (1.01 + 0.698i)T + (0.709 + 1.87i)T^{2} \) |
| 5 | \( 1 + (-2.41 + 2.72i)T + (-0.602 - 4.96i)T^{2} \) |
| 7 | \( 1 + (1.07 - 4.36i)T + (-6.19 - 3.25i)T^{2} \) |
| 11 | \( 1 + (2.95 - 2.03i)T + (3.90 - 10.2i)T^{2} \) |
| 17 | \( 1 + (-0.271 - 0.0668i)T + (15.0 + 7.90i)T^{2} \) |
| 19 | \( 1 - 6.22iT - 19T^{2} \) |
| 23 | \( 1 - 3.61T + 23T^{2} \) |
| 29 | \( 1 + (3.20 - 4.63i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (10.4 + 1.26i)T + (30.0 + 7.41i)T^{2} \) |
| 37 | \( 1 + (-2.68 - 0.325i)T + (35.9 + 8.85i)T^{2} \) |
| 41 | \( 1 + (-5.79 - 11.0i)T + (-23.2 + 33.7i)T^{2} \) |
| 43 | \( 1 + (-0.421 - 3.46i)T + (-41.7 + 10.2i)T^{2} \) |
| 47 | \( 1 + (0.944 + 0.358i)T + (35.1 + 31.1i)T^{2} \) |
| 53 | \( 1 + (-10.9 - 2.68i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (2.80 - 3.16i)T + (-7.11 - 58.5i)T^{2} \) |
| 61 | \( 1 + (6.51 - 1.60i)T + (54.0 - 28.3i)T^{2} \) |
| 67 | \( 1 + (-3.42 - 1.29i)T + (50.1 + 44.4i)T^{2} \) |
| 71 | \( 1 + (1.88 + 3.59i)T + (-40.3 + 58.4i)T^{2} \) |
| 73 | \( 1 + (-5.90 + 4.07i)T + (25.8 - 68.2i)T^{2} \) |
| 79 | \( 1 + (-1.18 + 3.11i)T + (-59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (-2.07 + 3.95i)T + (-47.1 - 68.3i)T^{2} \) |
| 89 | \( 1 - 2.14iT - 89T^{2} \) |
| 97 | \( 1 + (-6.50 - 7.33i)T + (-11.6 + 96.2i)T^{2} \) |
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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
−10.84081558911512265407260839569, −9.921533815273790855574720894588, −9.425799681643331680780517379185, −8.928593017281789600427989474527, −7.82420558810510627534438486088, −5.97632985203431575388584739437, −5.45192990809459931118136253389, −4.92276945226955280703821890851, −2.59312357844950791365186819790, −1.61835547169600789865968489689,
0.37478490231168583367420578260, 2.56650759192437816262042334595, 3.81477104528289825899483049285, 5.36636352868709142998095177321, 6.55465538247467156547405867914, 7.21101075634109763549749655292, 7.51394745580301833472018778512, 9.139530881825855774248130366762, 9.870275330114947861042890972715, 10.66258423949058563092532003386
