| L(s) = 1 | + (−0.173 − 0.984i)2-s + (−1.23 + 1.03i)3-s + (−0.939 + 0.342i)4-s + (−0.846 − 0.307i)5-s + (1.23 + 1.03i)6-s + (1.44 − 2.50i)7-s + (0.5 + 0.866i)8-s + (−0.0689 + 0.391i)9-s + (−0.156 + 0.886i)10-s + (0.5 + 0.866i)11-s + (0.806 − 1.39i)12-s + (−1.38 − 1.16i)13-s + (−2.71 − 0.989i)14-s + (1.36 − 0.496i)15-s + (0.766 − 0.642i)16-s + (−0.968 − 5.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.713 + 0.598i)3-s + (−0.469 + 0.171i)4-s + (−0.378 − 0.137i)5-s + (0.504 + 0.423i)6-s + (0.546 − 0.947i)7-s + (0.176 + 0.306i)8-s + (−0.0229 + 0.130i)9-s + (−0.0494 + 0.280i)10-s + (0.150 + 0.261i)11-s + (0.232 − 0.403i)12-s + (−0.384 − 0.323i)13-s + (−0.726 − 0.264i)14-s + (0.352 − 0.128i)15-s + (0.191 − 0.160i)16-s + (−0.234 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.245543 - 0.556546i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.245543 - 0.556546i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (1.25 + 4.17i)T \) |
| good | 3 | \( 1 + (1.23 - 1.03i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (0.846 + 0.307i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.44 + 2.50i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (1.38 + 1.16i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.968 + 5.49i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (5.45 - 1.98i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.00 + 5.69i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.05 + 5.29i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.84T + 37T^{2} \) |
| 41 | \( 1 + (-8.08 + 6.78i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (5.45 + 1.98i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.75 - 9.94i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (5.47 - 1.99i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.27 - 12.8i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.83 - 0.669i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.986 - 5.59i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (3.14 + 1.14i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-9.12 + 7.65i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.04 + 2.55i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (4.31 - 7.47i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.62 - 5.55i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (2.80 + 15.8i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.02492993025040495177561134927, −10.08785103889662896050604212591, −9.473121143900096736215106783350, −8.048301098343294185000493554961, −7.39693958107544842362447677911, −5.84977242078322946584701482001, −4.53378321700691735890645261259, −4.28889404621096631515745177968, −2.46289699261254620864482374314, −0.44936007552962713720040676275,
1.72263716199566822826590765707, 3.74884957448007850956026156214, 5.10608310420735065647547997246, 6.08445097479212787537515520063, 6.62119317852748147033237616822, 7.928391521756837326563312074369, 8.468620792848702888519074485962, 9.596695482803685159499520071288, 10.76017021224752579340969457413, 11.72730000618165505824975647662
