| L(s) = 1 | + (−5.86 − 12.1i)2-s + (−67.7 + 140. i)3-s + (45.7 − 57.3i)4-s + (−958. − 218. i)5-s + 2.11e3·6-s − 747. i·7-s + (−4.33e3 − 990. i)8-s + (−1.11e4 − 1.39e4i)9-s + (2.95e3 + 1.29e4i)10-s + (10.2 + 12.8i)11-s + (4.97e3 + 1.03e4i)12-s + (−1.94e3 + 8.52e3i)13-s + (−9.10e3 + 4.38e3i)14-s + (9.57e4 − 1.20e5i)15-s + (9.20e3 + 4.03e4i)16-s + (2.95e4 + 1.29e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.366 − 0.761i)2-s + (−0.837 + 1.73i)3-s + (0.178 − 0.223i)4-s + (−1.53 − 0.349i)5-s + 1.62·6-s − 0.311i·7-s + (−1.05 − 0.241i)8-s + (−1.69 − 2.12i)9-s + (0.295 + 1.29i)10-s + (0.000697 + 0.000874i)11-s + (0.239 + 0.497i)12-s + (−0.0681 + 0.298i)13-s + (−0.237 + 0.114i)14-s + (1.89 − 2.37i)15-s + (0.140 + 0.615i)16-s + (0.354 + 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0493i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.998 + 0.0493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.552660 - 0.0136508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.552660 - 0.0136508i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (3.03e6 - 1.57e6i)T \) |
| good | 2 | \( 1 + (5.86 + 12.1i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (67.7 - 140. i)T + (-4.09e3 - 5.12e3i)T^{2} \) |
| 5 | \( 1 + (958. + 218. i)T + (3.51e5 + 1.69e5i)T^{2} \) |
| 7 | \( 1 + 747. iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-10.2 - 12.8i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (1.94e3 - 8.52e3i)T + (-7.34e8 - 3.53e8i)T^{2} \) |
| 17 | \( 1 + (-2.95e4 - 1.29e5i)T + (-6.28e9 + 3.02e9i)T^{2} \) |
| 19 | \( 1 + (-3.89e4 - 3.10e4i)T + (3.77e9 + 1.65e10i)T^{2} \) |
| 23 | \( 1 + (1.11e5 + 1.39e5i)T + (-1.74e10 + 7.63e10i)T^{2} \) |
| 29 | \( 1 + (3.76e5 + 7.81e5i)T + (-3.11e11 + 3.91e11i)T^{2} \) |
| 31 | \( 1 + (-8.64e5 + 4.16e5i)T + (5.31e11 - 6.66e11i)T^{2} \) |
| 37 | \( 1 - 2.71e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (-2.80e6 + 1.35e6i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (-6.00e5 + 7.52e5i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (5.23e5 + 2.29e6i)T + (-5.60e13 + 2.70e13i)T^{2} \) |
| 59 | \( 1 + (-3.29e6 - 1.44e7i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-8.23e6 + 1.70e7i)T + (-1.19e14 - 1.49e14i)T^{2} \) |
| 67 | \( 1 + (-1.70e7 + 2.14e7i)T + (-9.03e13 - 3.95e14i)T^{2} \) |
| 71 | \( 1 + (2.92e7 + 2.33e7i)T + (1.43e14 + 6.29e14i)T^{2} \) |
| 73 | \( 1 + (1.88e7 + 4.29e6i)T + (7.26e14 + 3.49e14i)T^{2} \) |
| 79 | \( 1 - 3.00e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (6.13e7 + 2.95e7i)T + (1.40e15 + 1.76e15i)T^{2} \) |
| 89 | \( 1 + (4.77e6 - 9.90e6i)T + (-2.45e15 - 3.07e15i)T^{2} \) |
| 97 | \( 1 + (-3.28e7 - 4.11e7i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−14.87404122996401259119183847871, −12.16694213707982799676377208571, −11.55402699301217148689085513518, −10.63634946883396315234016438570, −9.781281901678669101475209650425, −8.393949874743666029149669765511, −6.07811005533435992235526624022, −4.42119235964438790905148207767, −3.52870084166272013770291026859, −0.56650639353969276097921245431,
0.55168886550727713694668748627, 2.82640530276372349390361523031, 5.55830403257972220737265767090, 7.09273378058745894869258834950, 7.40205174464174826223870410911, 8.478476564755252387877175298417, 11.31023677834221301052301801192, 11.83106737359999208392455792472, 12.67575217910881086307361804425, 14.27783916799248852208654533597
