| L(s) = 1 | + (−0.891 − 0.453i)2-s + (−0.873 + 1.49i)3-s + (0.587 + 0.809i)4-s + (−2.21 + 0.314i)5-s + (1.45 − 0.935i)6-s + (−2.72 − 2.72i)7-s + (−0.156 − 0.987i)8-s + (−1.47 − 2.61i)9-s + (2.11 + 0.725i)10-s + (0.335 − 0.109i)11-s + (−1.72 + 0.172i)12-s + (−1.12 − 2.20i)13-s + (1.19 + 3.66i)14-s + (1.46 − 3.58i)15-s + (−0.309 + 0.951i)16-s + (−3.49 + 0.554i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 − 0.321i)2-s + (−0.504 + 0.863i)3-s + (0.293 + 0.404i)4-s + (−0.990 + 0.140i)5-s + (0.594 − 0.382i)6-s + (−1.03 − 1.03i)7-s + (−0.0553 − 0.349i)8-s + (−0.491 − 0.871i)9-s + (0.668 + 0.229i)10-s + (0.101 − 0.0328i)11-s + (−0.497 + 0.0497i)12-s + (−0.312 − 0.612i)13-s + (0.318 + 0.980i)14-s + (0.378 − 0.925i)15-s + (−0.0772 + 0.237i)16-s + (−0.848 + 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00516079 - 0.0356955i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00516079 - 0.0356955i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.891 + 0.453i)T \) |
| 3 | \( 1 + (0.873 - 1.49i)T \) |
| 5 | \( 1 + (2.21 - 0.314i)T \) |
| good | 7 | \( 1 + (2.72 + 2.72i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.335 + 0.109i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.12 + 2.20i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (3.49 - 0.554i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (3.84 - 5.29i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (3.55 - 6.98i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-5.05 + 3.67i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.39 + 2.46i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.33 + 2.20i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (8.06 + 2.62i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (5.16 - 5.16i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.668 + 4.22i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-4.34 - 0.688i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (0.713 - 2.19i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.0451 - 0.139i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.18 - 7.47i)T + (-63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (3.62 + 4.98i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.30 + 4.74i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-0.803 - 1.10i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (0.915 + 5.77i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (0.633 + 1.94i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.93 - 1.09i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.23814299108347309777515853513, −11.36603907284106736249301729883, −10.36844593385915770728056763058, −9.876269666257262664829326151053, −8.509739737251639518531568762332, −7.33865399323986936302272066368, −6.16242927844901037530510236312, −4.21767449343086224362576091322, −3.42254086959155153570432080165, −0.04313656448321838951638068651,
2.52971131558187216425992012726, 4.81082117728733546414437466973, 6.45066218345151852667107271785, 6.89218552401673325865973815457, 8.379955800811090219419488312687, 8.994084628573038683847443401089, 10.54676636384370567797403140653, 11.62556113522040522498672799398, 12.31934039717655028610061861193, 13.16929722261356010945458052107
