| L(s) = 1 | + (4 + 6.92i)2-s + (−4.67 − 8.09i)3-s + (−31.9 + 55.4i)4-s + (−40.1 − 69.5i)5-s + (37.3 − 64.7i)6-s − 541.·7-s − 511.·8-s + (1.04e3 − 1.81e3i)9-s + (321. − 556. i)10-s − 6.30e3·11-s + 598.·12-s + (4.43e3 − 7.68e3i)13-s + (−2.16e3 − 3.75e3i)14-s + (−375. + 650. i)15-s + (−2.04e3 − 3.54e3i)16-s + (−1.69e4 − 2.93e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.0999 − 0.173i)3-s + (−0.249 + 0.433i)4-s + (−0.143 − 0.248i)5-s + (0.0706 − 0.122i)6-s − 0.597·7-s − 0.353·8-s + (0.480 − 0.831i)9-s + (0.101 − 0.175i)10-s − 1.42·11-s + 0.0999·12-s + (0.560 − 0.970i)13-s + (−0.211 − 0.365i)14-s + (−0.0287 + 0.0497i)15-s + (−0.125 − 0.216i)16-s + (−0.835 − 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.568575 - 0.655873i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.568575 - 0.655873i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 - 6.92i)T \) |
| 19 | \( 1 + (-2.10e3 - 2.98e4i)T \) |
| good | 3 | \( 1 + (4.67 + 8.09i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (40.1 + 69.5i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + 541.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.30e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-4.43e3 + 7.68e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.69e4 + 2.93e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-3.96e4 + 6.87e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.02e5 - 1.78e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 2.19e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.65e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-2.67e5 - 4.63e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-4.23e5 - 7.33e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (7.15e4 - 1.23e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (8.75e4 - 1.51e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (5.72e5 + 9.91e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.05e6 + 1.82e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.28e6 + 2.22e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.55e6 + 2.68e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.53e6 + 2.65e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.46e6 - 4.27e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 - 7.31e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-3.41e6 + 5.92e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-2.97e6 - 5.16e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−14.58149366637265615071951483616, −12.99672999141249580013441622757, −12.62594613974053609824698677531, −10.76276609010330054775798092351, −9.229400233654460747599426207673, −7.77573922583502466297771015402, −6.46965473045715781424831775448, −5.01001210911286402326677470124, −3.16888310036975452016936884718, −0.32972433904153613300330676778,
2.11318374335025961698933099866, 3.88220364418299905856719058912, 5.46037362611668318756954733701, 7.24378801719394721497092304790, 9.056570065522515980519003003883, 10.52327652757960675488760165200, 11.21290905288060123111690155109, 13.03111337261605836647367298566, 13.41903024735717664212870224010, 15.21421271581293161688091570793
