L(s) = 1 | − 8.23i·2-s + (−13.6 − 7.44i)3-s − 35.8·4-s + 95.6·5-s + (−61.3 + 112. i)6-s + 31.5i·8-s + (132. + 203. i)9-s − 788. i·10-s − 61.6i·11-s + (490. + 266. i)12-s − 701. i·13-s + (−1.31e3 − 712. i)15-s − 886.·16-s − 2.09e3·17-s + (1.67e3 − 1.08e3i)18-s − 766. i·19-s + ⋯ |
L(s) = 1 | − 1.45i·2-s + (−0.878 − 0.477i)3-s − 1.11·4-s + 1.71·5-s + (−0.695 + 1.27i)6-s + 0.174i·8-s + (0.543 + 0.839i)9-s − 2.49i·10-s − 0.153i·11-s + (0.983 + 0.534i)12-s − 1.15i·13-s + (−1.50 − 0.817i)15-s − 0.865·16-s − 1.76·17-s + (1.22 − 0.791i)18-s − 0.487i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.307534831\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.307534831\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (13.6 + 7.44i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 8.23iT - 32T^{2} \) |
| 5 | \( 1 - 95.6T + 3.12e3T^{2} \) |
| 11 | \( 1 + 61.6iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 701. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 766. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.16e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.79e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.89e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 4.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.78e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.91e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.36e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.16e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 7.81e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.03e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.74e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 1.69e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.48e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.50e4iT - 8.58e9T^{2} \) |
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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
−11.31784411007455083701329303905, −10.61157428589420917198320993072, −9.922230168556191708253961663567, −8.813373417055501737622319201323, −6.85580807504737844167222377529, −5.85914730783978257567550162196, −4.69080881020977081621641161153, −2.64979081338984729923770262516, −1.75381228134521014871905337419, −0.46997541488758973267669141706,
1.89123775691043203846403637851, 4.51856003824736790721437455748, 5.43844604047364818090959968132, 6.40338169525904363783615319006, 6.86085409021983988678622660386, 8.784486237610092739682400946557, 9.497883811990144270726362778385, 10.55470030733810146077544285529, 11.73002687075144707296031727354, 13.23011096167374554781379009508