| L(s) = 1 | − 13.4i·2-s + (−34.2 + 19.7i)3-s − 116.·4-s + (−28.7 + 16.6i)5-s + (265. + 459. i)6-s + (−186. − 107. i)7-s + 704. i·8-s + (415. − 720. i)9-s + (223. + 386. i)10-s + 2.28e3·11-s + (3.98e3 − 2.29e3i)12-s + (−344. + 596. i)13-s + (−1.44e3 + 2.50e3i)14-s + (656. − 1.13e3i)15-s + 2.00e3·16-s + (−510. + 883. i)17-s + ⋯ |
| L(s) = 1 | − 1.67i·2-s + (−1.26 + 0.731i)3-s − 1.81·4-s + (−0.230 + 0.132i)5-s + (1.22 + 2.12i)6-s + (−0.543 − 0.313i)7-s + 1.37i·8-s + (0.570 − 0.988i)9-s + (0.223 + 0.386i)10-s + 1.71·11-s + (2.30 − 1.33i)12-s + (−0.156 + 0.271i)13-s + (−0.526 + 0.912i)14-s + (0.194 − 0.336i)15-s + 0.490·16-s + (−0.103 + 0.179i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.732490 - 0.171159i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.732490 - 0.171159i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.12e4 - 6.79e4i)T \) |
| good | 2 | \( 1 + 13.4iT - 64T^{2} \) |
| 3 | \( 1 + (34.2 - 19.7i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (28.7 - 16.6i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (186. + 107. i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 - 2.28e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (344. - 596. i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + (510. - 883. i)T + (-1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-461. + 266. i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-4.19e3 - 7.27e3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-3.28e4 - 1.89e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-1.20e4 - 2.09e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-7.24e4 + 4.18e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 - 6.00e4T + 4.75e9T^{2} \) |
| 47 | \( 1 + 1.84e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.23e4 - 2.14e4i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + 3.37e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + (-6.32e4 - 3.65e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.37e5 - 4.11e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-1.76e5 - 1.02e5i)T + (6.40e10 + 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-1.85e5 - 1.07e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (5.51e4 - 9.55e4i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.83e5 + 3.17e5i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-1.00e6 + 5.82e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 5.97e5T + 8.32e11T^{2} \) |
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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
−14.35069382136714754961646653650, −12.85856393340192292706016081688, −11.72689471875975129019734505818, −11.25319834906848320412940258798, −10.08005454960811120056099506395, −9.242158480932789265591931948433, −6.50107022808918768219068985100, −4.62738784990759466196827350455, −3.53213172621964049449780235903, −1.07942562087223403732503570781,
0.54889944266192950816339370142, 4.62635068497229074795960608920, 6.25130085516614514371215424110, 6.51807890825023024897681489431, 8.024931947085952391110759961144, 9.490241733717224203888229254113, 11.55093553281382543837288834225, 12.49932733182126689503245140456, 13.81234030694059236183950481973, 15.03064129246375493171632608495
