L(s) = 1 | + (−1.77 + 3.06i)2-s + (1.5 − 0.866i)3-s + (−4.27 − 7.40i)4-s + (−7.27 − 4.20i)5-s + 6.13i·6-s + (6.51 + 2.55i)7-s + 16.1·8-s + (1.5 − 2.59i)9-s + (25.7 − 14.8i)10-s + (1.65 + 2.87i)11-s + (−12.8 − 7.40i)12-s + 19.6i·13-s + (−19.3 + 15.4i)14-s − 14.5·15-s + (−11.4 + 19.8i)16-s + (−2.81 + 1.62i)17-s + ⋯ |
L(s) = 1 | + (−0.885 + 1.53i)2-s + (0.5 − 0.288i)3-s + (−1.06 − 1.85i)4-s + (−1.45 − 0.840i)5-s + 1.02i·6-s + (0.930 + 0.365i)7-s + 2.01·8-s + (0.166 − 0.288i)9-s + (2.57 − 1.48i)10-s + (0.150 + 0.261i)11-s + (−1.06 − 0.616i)12-s + 1.51i·13-s + (−1.38 + 1.10i)14-s − 0.970·15-s + (−0.714 + 1.23i)16-s + (−0.165 + 0.0957i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.414551 + 0.737629i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.414551 + 0.737629i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 + (-6.51 - 2.55i)T \) |
| 11 | \( 1 + (-1.65 - 2.87i)T \) |
good | 2 | \( 1 + (1.77 - 3.06i)T + (-2 - 3.46i)T^{2} \) |
| 5 | \( 1 + (7.27 + 4.20i)T + (12.5 + 21.6i)T^{2} \) |
| 13 | \( 1 - 19.6iT - 169T^{2} \) |
| 17 | \( 1 + (2.81 - 1.62i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-21.3 - 12.3i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (10.4 - 18.1i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 35.3T + 841T^{2} \) |
| 31 | \( 1 + (-22.8 + 13.1i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-10.1 + 17.5i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 44.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 63.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-12.8 - 7.44i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (4.45 + 7.71i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-45.7 + 26.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-68.6 - 39.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (46.9 + 81.4i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 25.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (88.8 - 51.3i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (2.01 - 3.48i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 97.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-127. - 73.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 90.9iT - 9.40e3T^{2} \) |
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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
−12.04053022548973576051411069729, −11.57070458108237127972925488710, −9.746889235174002132509156940565, −8.884904783362029433601033524705, −8.147651300981662341487176468958, −7.66853460767970839579100117715, −6.63478524683912676572906496474, −5.12473854278750224679476694123, −4.16883092612728459779394798782, −1.28293489155018023547741885843,
0.70234920492261493722167757072, 2.75301979688444814597005554693, 3.50975857947693208486611290987, 4.64802079684678571700265811008, 7.25682480033290288501591451621, 8.128826484568346947176167114124, 8.590187959913182328331635333193, 10.21118054014524663285028887571, 10.55642300065825055094874639894, 11.56315168568978745387052519805