| L(s) = 1 | + (0.991 + 0.128i)2-s + (−0.225 + 0.331i)3-s + (0.967 + 0.254i)4-s + (0.00709 + 0.771i)5-s + (−0.265 + 0.299i)6-s + (−0.345 − 0.163i)7-s + (0.926 + 0.376i)8-s + (1.04 + 2.64i)9-s + (−0.0919 + 0.766i)10-s + (−3.45 − 2.12i)11-s + (−0.302 + 0.263i)12-s + (0.549 + 5.41i)13-s + (−0.322 − 0.206i)14-s + (−0.257 − 0.171i)15-s + (0.870 + 0.492i)16-s + (4.23 + 4.27i)17-s + ⋯ |
| L(s) = 1 | + (0.701 + 0.0906i)2-s + (−0.130 + 0.191i)3-s + (0.483 + 0.127i)4-s + (0.00317 + 0.345i)5-s + (−0.108 + 0.122i)6-s + (−0.130 − 0.0616i)7-s + (0.327 + 0.133i)8-s + (0.348 + 0.880i)9-s + (−0.0290 + 0.242i)10-s + (−1.04 − 0.640i)11-s + (−0.0872 + 0.0759i)12-s + (0.152 + 1.50i)13-s + (−0.0860 − 0.0551i)14-s + (−0.0664 − 0.0442i)15-s + (0.217 + 0.123i)16-s + (1.02 + 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61316 + 1.32013i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61316 + 1.32013i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.991 - 0.128i)T \) |
| 19 | \( 1 + (4.29 - 0.720i)T \) |
| good | 3 | \( 1 + (0.225 - 0.331i)T + (-1.10 - 2.78i)T^{2} \) |
| 5 | \( 1 + (-0.00709 - 0.771i)T + (-4.99 + 0.0918i)T^{2} \) |
| 7 | \( 1 + (0.345 + 0.163i)T + (4.45 + 5.40i)T^{2} \) |
| 11 | \( 1 + (3.45 + 2.12i)T + (4.96 + 9.81i)T^{2} \) |
| 13 | \( 1 + (-0.549 - 5.41i)T + (-12.7 + 2.60i)T^{2} \) |
| 17 | \( 1 + (-4.23 - 4.27i)T + (-0.156 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-6.89 + 0.507i)T + (22.7 - 3.36i)T^{2} \) |
| 29 | \( 1 + (0.283 - 0.401i)T + (-9.66 - 27.3i)T^{2} \) |
| 31 | \( 1 + (2.14 + 6.86i)T + (-25.4 + 17.6i)T^{2} \) |
| 37 | \( 1 + (5.35 - 2.89i)T + (20.2 - 30.9i)T^{2} \) |
| 41 | \( 1 + (1.49 - 1.56i)T + (-1.88 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-7.24 - 3.10i)T + (29.6 + 31.0i)T^{2} \) |
| 47 | \( 1 + (4.34 + 0.889i)T + (43.2 + 18.4i)T^{2} \) |
| 53 | \( 1 + (-1.75 - 5.27i)T + (-42.4 + 31.7i)T^{2} \) |
| 59 | \( 1 + (-2.04 - 6.98i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-5.13 + 4.63i)T + (6.15 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-3.42 - 0.125i)T + (66.8 + 4.91i)T^{2} \) |
| 71 | \( 1 + (1.77 + 1.60i)T + (7.16 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-3.46 + 12.7i)T + (-62.8 - 37.0i)T^{2} \) |
| 79 | \( 1 + (-9.24 + 6.93i)T + (22.1 - 75.8i)T^{2} \) |
| 83 | \( 1 + (0.465 - 1.24i)T + (-62.5 - 54.5i)T^{2} \) |
| 89 | \( 1 + (4.58 + 16.7i)T + (-76.6 + 45.2i)T^{2} \) |
| 97 | \( 1 + (-4.59 + 0.168i)T + (96.7 - 7.12i)T^{2} \) |
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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
−10.82140401006156808557483533439, −10.01856200940984274197608224070, −8.761077510378371175028909858896, −7.87726564474259378925812478172, −6.96432226526216264833342198630, −6.08009780116570838246728325886, −5.12090086071483226825557139588, −4.23992575311986467291915958415, −3.12484290046299552547008926986, −1.89526110134630000059451422199,
0.912362420494922413814311751823, 2.68111667076306182542602817164, 3.56684690209225664972300159569, 5.04154503014109005236313866094, 5.38744537402255243835770144678, 6.72685616203885166273119471156, 7.37484325730319828124386665955, 8.429937462499006319544502636632, 9.491518734880506420261100246191, 10.37816436555864995374976052863
