| L(s) = 1 | − 42.7·2-s + (18.2 − 31.6i)3-s + 1.31e3·4-s + (−1.01e3 + 1.74e3i)5-s + (−781. + 1.35e3i)6-s + (−3.15e3 − 5.46e3i)7-s − 3.43e4·8-s + (9.17e3 + 1.58e4i)9-s + (4.31e4 − 7.47e4i)10-s − 4.85e4·11-s + (2.40e4 − 4.16e4i)12-s + (6.69e4 + 1.16e5i)13-s + (1.34e5 + 2.33e5i)14-s + (3.69e4 + 6.39e4i)15-s + 7.92e5·16-s + (2.35e5 + 4.07e5i)17-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + (0.130 − 0.225i)3-s + 2.56·4-s + (−0.722 + 1.25i)5-s + (−0.246 + 0.426i)6-s + (−0.496 − 0.860i)7-s − 2.96·8-s + (0.466 + 0.807i)9-s + (1.36 − 2.36i)10-s − 0.999·11-s + (0.334 − 0.579i)12-s + (0.650 + 1.12i)13-s + (0.938 + 1.62i)14-s + (0.188 + 0.326i)15-s + 3.02·16-s + (0.683 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.623i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0201552 - 0.0575768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0201552 - 0.0575768i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.46e6 + 2.21e7i)T \) |
| good | 2 | \( 1 + 42.7T + 512T^{2} \) |
| 3 | \( 1 + (-18.2 + 31.6i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.01e3 - 1.74e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (3.15e3 + 5.46e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + 4.85e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-6.69e4 - 1.16e5i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-2.35e5 - 4.07e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.30e5 + 3.99e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (2.60e5 - 4.51e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (7.27e5 + 1.25e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-784. + 1.35e3i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (1.02e7 - 1.78e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + 2.46e7T + 3.27e14T^{2} \) |
| 47 | \( 1 + 1.76e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (-3.27e7 + 5.67e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 - 1.09e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + (1.00e8 + 1.73e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.27e8 - 2.20e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (1.68e8 + 2.92e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-8.85e7 - 1.53e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.20e8 + 3.81e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-1.11e8 + 1.93e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (8.00e7 - 1.38e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 6.11e8T + 7.60e17T^{2} \) |
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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
−15.23005256131323234395552289829, −13.55533497088367331044687571100, −11.64220063526751657476800332960, −10.61478809632383152532137328767, −10.10999987681868197820018281819, −8.312246276307691837705551688809, −7.38004036329272050714730161330, −6.67303532212876809525006901403, −3.31132737381688245240849213865, −1.73658963456677686073349907717,
0.04620910831777589798533335129, 1.06700802162727663952454183826, 3.07802376807202240851381530760, 5.64658528659331708861523811200, 7.51025318125356933662923326157, 8.493109517381762318506459400969, 9.318822037248438532307998255436, 10.36690614448284727697445151333, 11.93858779072057758187231399188, 12.62633574822509942417107481537
