| L(s) = 1 | + (2.13 − 0.828i)2-s + (−0.750 − 1.56i)3-s + (2.41 − 2.19i)4-s + (−0.931 + 3.27i)5-s + (−2.89 − 2.71i)6-s + (−1.78 + 2.70i)7-s + (1.29 − 2.59i)8-s + (−1.87 + 2.34i)9-s + (0.720 + 7.77i)10-s + (4.53 + 3.64i)11-s + (−5.24 − 2.11i)12-s + (−0.793 + 1.19i)13-s + (−1.59 + 7.26i)14-s + (5.81 − 1.00i)15-s + (0.0116 − 0.126i)16-s + (−6.62 + 3.55i)17-s + ⋯ |
| L(s) = 1 | + (1.51 − 0.586i)2-s + (−0.433 − 0.901i)3-s + (1.20 − 1.09i)4-s + (−0.416 + 1.46i)5-s + (−1.18 − 1.10i)6-s + (−0.676 + 1.02i)7-s + (0.457 − 0.918i)8-s + (−0.624 + 0.780i)9-s + (0.227 + 2.46i)10-s + (1.36 + 1.10i)11-s + (−1.51 − 0.610i)12-s + (−0.219 + 0.331i)13-s + (−0.425 + 1.94i)14-s + (1.50 − 0.258i)15-s + (0.00292 − 0.0315i)16-s + (−1.60 + 0.863i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.19421 + 0.787953i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.19421 + 0.787953i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.750 + 1.56i)T \) |
| 103 | \( 1 + (7.11 - 7.23i)T \) |
| good | 2 | \( 1 + (-2.13 + 0.828i)T + (1.47 - 1.34i)T^{2} \) |
| 5 | \( 1 + (0.931 - 3.27i)T + (-4.25 - 2.63i)T^{2} \) |
| 7 | \( 1 + (1.78 - 2.70i)T + (-2.72 - 6.44i)T^{2} \) |
| 11 | \( 1 + (-4.53 - 3.64i)T + (2.35 + 10.7i)T^{2} \) |
| 13 | \( 1 + (0.793 - 1.19i)T + (-5.06 - 11.9i)T^{2} \) |
| 17 | \( 1 + (6.62 - 3.55i)T + (9.39 - 14.1i)T^{2} \) |
| 19 | \( 1 + (1.07 + 3.04i)T + (-14.8 + 11.9i)T^{2} \) |
| 23 | \( 1 + (1.00 + 6.49i)T + (-21.9 + 6.97i)T^{2} \) |
| 29 | \( 1 + (-0.813 + 2.85i)T + (-24.6 - 15.2i)T^{2} \) |
| 31 | \( 1 + (-3.87 + 1.78i)T + (20.1 - 23.5i)T^{2} \) |
| 37 | \( 1 + (-1.44 + 1.91i)T + (-10.1 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-1.86 - 0.468i)T + (36.1 + 19.3i)T^{2} \) |
| 43 | \( 1 + (3.01 + 3.99i)T + (-11.7 + 41.3i)T^{2} \) |
| 47 | \( 1 + (-3.15 - 5.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.71 - 4.33i)T + (-8.12 - 52.3i)T^{2} \) |
| 59 | \( 1 + (-1.84 - 2.79i)T + (-22.9 + 54.3i)T^{2} \) |
| 61 | \( 1 + (-7.42 + 3.98i)T + (33.6 - 50.8i)T^{2} \) |
| 67 | \( 1 + (-7.15 + 14.3i)T + (-40.3 - 53.4i)T^{2} \) |
| 71 | \( 1 + (-12.7 - 3.19i)T + (62.5 + 33.5i)T^{2} \) |
| 73 | \( 1 + (-3.15 - 11.0i)T + (-62.0 + 38.4i)T^{2} \) |
| 79 | \( 1 + (0.395 + 0.407i)T + (-2.43 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-2.68 - 5.38i)T + (-50.0 + 66.2i)T^{2} \) |
| 89 | \( 1 + (-8.02 - 7.31i)T + (8.21 + 88.6i)T^{2} \) |
| 97 | \( 1 + (7.10 - 4.39i)T + (43.2 - 86.8i)T^{2} \) |
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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
−10.80814954323839969313080601946, −9.532051856223577052443316352298, −8.387166328723574968195142209625, −6.87287097227027322223927584518, −6.57765354157332693647897812953, −6.11503613227315708012146887616, −4.67269296059798370687107744729, −3.86869510192349877721428746588, −2.42993507930437132651965374452, −2.29675039735490865627350014405,
0.68720517026945103556813746438, 3.39077553961644547598103065344, 3.93805386320126544178245906329, 4.60869075741999576270774344821, 5.38029028793505905235774683949, 6.29805096447725513061846361002, 6.97814018975177427205813545758, 8.315955164076541473762523329354, 9.144597272396393014606692497447, 9.895409913354453322397418035578
