L(s) = 1 | + (−0.278 − 1.70i)3-s + (2.42 − 0.884i)5-s + (−0.245 − 1.39i)7-s + (−2.84 + 0.950i)9-s + (−1.99 − 0.725i)11-s + (1.59 + 1.33i)13-s + (−2.18 − 3.90i)15-s + (1.93 − 3.35i)17-s + (1.22 + 2.11i)19-s + (−2.31 + 0.807i)21-s + (1.45 − 8.24i)23-s + (1.29 − 1.08i)25-s + (2.41 + 4.59i)27-s + (−0.797 + 0.669i)29-s + (−1.54 + 8.73i)31-s + ⋯ |
L(s) = 1 | + (−0.160 − 0.987i)3-s + (1.08 − 0.395i)5-s + (−0.0928 − 0.526i)7-s + (−0.948 + 0.316i)9-s + (−0.600 − 0.218i)11-s + (0.441 + 0.370i)13-s + (−0.564 − 1.00i)15-s + (0.469 − 0.813i)17-s + (0.280 + 0.485i)19-s + (−0.505 + 0.176i)21-s + (0.303 − 1.71i)23-s + (0.258 − 0.216i)25-s + (0.465 + 0.885i)27-s + (−0.148 + 0.124i)29-s + (−0.276 + 1.56i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.987013 - 0.812857i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.987013 - 0.812857i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.278 + 1.70i)T \) |
good | 5 | \( 1 + (-2.42 + 0.884i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.245 + 1.39i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.99 + 0.725i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.59 - 1.33i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.93 + 3.35i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.22 - 2.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.45 + 8.24i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.797 - 0.669i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.54 - 8.73i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (4.97 - 8.60i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.25 - 7.76i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.70 - 2.07i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.665 - 3.77i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 3.73T + 53T^{2} \) |
| 59 | \( 1 + (5.04 - 1.83i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.85 + 10.5i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-11.3 - 9.50i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.68 + 4.64i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.25 + 9.09i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.10 - 5.12i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (9.32 - 7.82i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (7.20 + 12.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.6 - 3.87i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.39833590689915845751812199938, −11.15395413373494791646304250414, −10.22512115749939272436034870420, −9.082786241047342687437884704666, −8.065583425756508978622469064013, −6.91262335261889162783682033484, −5.99031159685522276923130098203, −4.93526439911789279830465735020, −2.83578194242709220346791019291, −1.27409770171001541349826215318,
2.42003864231724442848438475955, 3.81600176385302263931102299341, 5.59827354394762921707113437763, 5.77710640992093789961344817479, 7.53678880356792319546464227869, 8.966695480511274345815668310653, 9.653214487524999569480773455550, 10.51975762969910362130611212243, 11.27520449368013059208121193243, 12.55672207762687500323342132718