| L(s) = 1 | + (−2.62 − 5.45i)2-s + (−13.6 + 1.02i)3-s + (17.0 − 21.4i)4-s + (29.1 + 94.5i)5-s + (41.3 + 71.6i)6-s + (−270. − 156. i)7-s + (−539. − 123. i)8-s + (−535. + 80.7i)9-s + (438. − 407. i)10-s + (1.16e3 + 1.46e3i)11-s + (−210. + 309. i)12-s + (2.31e3 + 2.14e3i)13-s + (−141. + 1.88e3i)14-s + (−494. − 1.25e3i)15-s + (354. + 1.55e3i)16-s + (308. + 95.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.328 − 0.681i)2-s + (−0.505 + 0.0378i)3-s + (0.266 − 0.334i)4-s + (0.233 + 0.756i)5-s + (0.191 + 0.331i)6-s + (−0.789 − 0.455i)7-s + (−1.05 − 0.240i)8-s + (−0.735 + 0.110i)9-s + (0.438 − 0.407i)10-s + (0.874 + 1.09i)11-s + (−0.122 + 0.179i)12-s + (1.05 + 0.978i)13-s + (−0.0515 + 0.687i)14-s + (−0.146 − 0.373i)15-s + (0.0865 + 0.379i)16-s + (0.0628 + 0.0193i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.623061 + 0.392768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.623061 + 0.392768i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.01e4 + 5.19e4i)T \) |
| good | 2 | \( 1 + (2.62 + 5.45i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (13.6 - 1.02i)T + (720. - 108. i)T^{2} \) |
| 5 | \( 1 + (-29.1 - 94.5i)T + (-1.29e4 + 8.80e3i)T^{2} \) |
| 7 | \( 1 + (270. + 156. i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.16e3 - 1.46e3i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-2.31e3 - 2.14e3i)T + (3.60e5 + 4.81e6i)T^{2} \) |
| 17 | \( 1 + (-308. - 95.2i)T + (1.99e7 + 1.35e7i)T^{2} \) |
| 19 | \( 1 + (472. - 3.13e3i)T + (-4.49e7 - 1.38e7i)T^{2} \) |
| 23 | \( 1 + (8.27e3 - 2.10e4i)T + (-1.08e8 - 1.00e8i)T^{2} \) |
| 29 | \( 1 + (3.20e4 + 2.40e3i)T + (5.88e8 + 8.86e7i)T^{2} \) |
| 31 | \( 1 + (-9.17e3 - 6.25e3i)T + (3.24e8 + 8.26e8i)T^{2} \) |
| 37 | \( 1 + (5.14e4 - 2.96e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-6.26e4 + 3.01e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (7.82e4 - 9.81e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-6.10e4 + 5.66e4i)T + (1.65e9 - 2.21e10i)T^{2} \) |
| 59 | \( 1 + (-6.36e4 - 2.78e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (1.23e5 + 1.81e5i)T + (-1.88e10 + 4.79e10i)T^{2} \) |
| 67 | \( 1 + (5.84e5 + 8.81e4i)T + (8.64e10 + 2.66e10i)T^{2} \) |
| 71 | \( 1 + (-4.91e5 + 1.92e5i)T + (9.39e10 - 8.71e10i)T^{2} \) |
| 73 | \( 1 + (1.53e5 - 1.65e5i)T + (-1.13e10 - 1.50e11i)T^{2} \) |
| 79 | \( 1 + (3.27e5 - 5.67e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.38e4 + 1.84e5i)T + (-3.23e11 + 4.87e10i)T^{2} \) |
| 89 | \( 1 + (-1.03e6 + 7.73e4i)T + (4.91e11 - 7.40e10i)T^{2} \) |
| 97 | \( 1 + (-2.37e5 - 2.98e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.84228085501397826628083643034, −13.77002774655438516056966906417, −12.06639933337775427491086084079, −11.24986979200090413055685064864, −10.19974396530018730356131596228, −9.234233234868569620428160979634, −6.88132497007361416650133704606, −5.99886117025582799701152093797, −3.54228341808656194714471004349, −1.69097140227707049782462189852,
0.40219567369908096407807228867, 3.21104153333677076556620597981, 5.74196911495942522509829958717, 6.39921793561428520612199536786, 8.403423592389862041148660094639, 9.024818565645114701713959978806, 11.02217014393963769466385451895, 12.14616852268410603960528113441, 13.15659379812050729353682285170, 14.75228874478808721030211976313
