| L(s) = 1 | + (0.758 − 1.57i)2-s + (6.18 − 9.07i)3-s + (37.9 + 47.6i)4-s + (−99.6 − 107. i)5-s + (−9.60 − 16.6i)6-s + (−494. − 285. i)7-s + (213. − 48.6i)8-s + (222. + 566. i)9-s + (−244. + 75.5i)10-s + (1.48e3 − 1.85e3i)11-s + (667. − 49.9i)12-s + (−3.55e3 − 1.09e3i)13-s + (−824. + 562. i)14-s + (−1.59e3 + 239. i)15-s + (−782. + 3.42e3i)16-s + (−5.51e3 − 5.11e3i)17-s + ⋯ |
| L(s) = 1 | + (0.0948 − 0.197i)2-s + (0.229 − 0.335i)3-s + (0.593 + 0.744i)4-s + (−0.797 − 0.859i)5-s + (−0.0444 − 0.0769i)6-s + (−1.44 − 0.831i)7-s + (0.416 − 0.0949i)8-s + (0.304 + 0.776i)9-s + (−0.244 + 0.0755i)10-s + (1.11 − 1.39i)11-s + (0.386 − 0.0289i)12-s + (−1.61 − 0.498i)13-s + (−0.300 + 0.204i)14-s + (−0.471 + 0.0710i)15-s + (−0.191 + 0.837i)16-s + (−1.12 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.383031 - 1.04404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.383031 - 1.04404i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.80e4 + 4.10e4i)T \) |
| good | 2 | \( 1 + (-0.758 + 1.57i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (-6.18 + 9.07i)T + (-266. - 678. i)T^{2} \) |
| 5 | \( 1 + (99.6 + 107. i)T + (-1.16e3 + 1.55e4i)T^{2} \) |
| 7 | \( 1 + (494. + 285. i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.48e3 + 1.85e3i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (3.55e3 + 1.09e3i)T + (3.98e6 + 2.71e6i)T^{2} \) |
| 17 | \( 1 + (5.51e3 + 5.11e3i)T + (1.80e6 + 2.40e7i)T^{2} \) |
| 19 | \( 1 + (789. + 309. i)T + (3.44e7 + 3.19e7i)T^{2} \) |
| 23 | \( 1 + (-789. - 119. i)T + (1.41e8 + 4.36e7i)T^{2} \) |
| 29 | \( 1 + (-1.02e4 - 1.50e4i)T + (-2.17e8 + 5.53e8i)T^{2} \) |
| 31 | \( 1 + (-11.1 - 149. i)T + (-8.77e8 + 1.32e8i)T^{2} \) |
| 37 | \( 1 + (-4.54e4 + 2.62e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (3.55e4 + 1.70e4i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-8.99e4 - 1.12e5i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (1.21e5 - 3.74e4i)T + (1.83e10 - 1.24e10i)T^{2} \) |
| 59 | \( 1 + (-1.20e4 + 5.29e4i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.01e5 - 7.59e3i)T + (5.09e10 + 7.67e9i)T^{2} \) |
| 67 | \( 1 + (2.23e4 - 5.69e4i)T + (-6.63e10 - 6.15e10i)T^{2} \) |
| 71 | \( 1 + (5.95e4 + 3.94e5i)T + (-1.22e11 + 3.77e10i)T^{2} \) |
| 73 | \( 1 + (-1.67e5 + 5.42e5i)T + (-1.25e11 - 8.52e10i)T^{2} \) |
| 79 | \( 1 + (-1.71e4 + 2.96e4i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.61e5 + 1.09e5i)T + (1.19e11 + 3.04e11i)T^{2} \) |
| 89 | \( 1 + (-1.09e5 + 1.60e5i)T + (-1.81e11 - 4.62e11i)T^{2} \) |
| 97 | \( 1 + (1.02e5 - 1.28e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−13.86002300861000538973603514979, −12.91081465517838041751996247281, −12.10972863762675195641960226238, −10.82072133946858994891716214423, −9.123080749145224451472357639982, −7.72465278562442355195140010878, −6.78470905982029530360139192341, −4.31962293719793786413122226038, −2.91288442704404054699071510203, −0.46736456096691417960652749547,
2.45715611178429754391929221673, 4.18693390862446265231392103060, 6.49192568007226154643855093895, 6.97989195858620798750322737660, 9.390822245979497555019436888438, 10.03706002413056931293100864532, 11.70098447072510614716271060651, 12.53631813668173795494677781908, 14.71222492899808242878160138994, 15.05388641411448588011944866343
