| L(s) = 1 | − 6.04·2-s + (46.4 + 80.4i)3-s − 91.4·4-s + (84.4 + 146. i)5-s + (−280. − 486. i)6-s + (−519. + 899. i)7-s + 1.32e3·8-s + (−3.22e3 + 5.58e3i)9-s + (−510. − 883. i)10-s + 2.07e3·11-s + (−4.24e3 − 7.35e3i)12-s + (6.83e3 − 1.18e4i)13-s + (3.13e3 − 5.43e3i)14-s + (−7.84e3 + 1.35e4i)15-s + 3.68e3·16-s + (−1.19e4 + 2.07e4i)17-s + ⋯ |
| L(s) = 1 | − 0.534·2-s + (0.993 + 1.72i)3-s − 0.714·4-s + (0.301 + 0.523i)5-s + (−0.530 − 0.919i)6-s + (−0.572 + 0.990i)7-s + 0.916·8-s + (−1.47 + 2.55i)9-s + (−0.161 − 0.279i)10-s + 0.469·11-s + (−0.709 − 1.22i)12-s + (0.862 − 1.49i)13-s + (0.305 − 0.529i)14-s + (−0.600 + 1.03i)15-s + 0.224·16-s + (−0.590 + 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.133678 - 1.26807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.133678 - 1.26807i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.49e5 + 2.64e5i)T \) |
| good | 2 | \( 1 + 6.04T + 128T^{2} \) |
| 3 | \( 1 + (-46.4 - 80.4i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-84.4 - 146. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (519. - 899. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 2.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-6.83e3 + 1.18e4i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.19e4 - 2.07e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (5.44e3 + 9.42e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.81e4 + 3.13e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (2.49e4 - 4.32e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-6.51e4 - 1.12e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.03e4 - 3.52e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 2.88e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 6.11e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-7.11e5 - 1.23e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 - 1.19e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (3.76e5 - 6.52e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-6.54e5 - 1.13e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.24e6 - 3.88e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (6.91e5 - 1.19e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (2.99e6 - 5.19e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-8.79e5 - 1.52e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (2.21e6 + 3.82e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.33e7T + 8.07e13T^{2} \) |
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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
−15.16559445238563219312753385810, −14.22407336419434169265394978257, −13.05215302433925280274605615464, −10.68548938118567644357933413198, −10.11079960913751559164527452381, −8.879112298921712816727647706416, −8.419008624145106554239907831918, −5.65008211134078292234193881528, −4.06526354274441943957589519210, −2.77291673690058444035829093024,
0.60165650623552499785219406173, 1.67187697038059964357086606408, 3.85822027208186051353972660865, 6.50615781780927177289212583974, 7.54928480007067677992436353990, 8.845856889418195511682743872851, 9.452623399508678861297897642269, 11.69384195797205024329581738462, 13.25262850583187160602304115043, 13.48579738150589633890116266705
