L(s) = 1 | + (3.25 + 4.62i)2-s − 1.29·3-s + (−10.8 + 30.1i)4-s + (−51.3 − 21.9i)5-s + (−4.22 − 6.00i)6-s + 170. i·7-s + (−174. + 47.7i)8-s − 241.·9-s + (−65.4 − 309. i)10-s − 39.8i·11-s + (14.0 − 39.0i)12-s + 537.·13-s + (−788. + 554. i)14-s + (66.7 + 28.5i)15-s + (−789. − 652. i)16-s + 1.35e3i·17-s + ⋯ |
L(s) = 1 | + (0.575 + 0.818i)2-s − 0.0832·3-s + (−0.338 + 0.940i)4-s + (−0.919 − 0.393i)5-s + (−0.0478 − 0.0681i)6-s + 1.31i·7-s + (−0.964 + 0.264i)8-s − 0.993·9-s + (−0.206 − 0.978i)10-s − 0.0992i·11-s + (0.0282 − 0.0783i)12-s + 0.881·13-s + (−1.07 + 0.755i)14-s + (0.0765 + 0.0327i)15-s + (−0.770 − 0.637i)16-s + 1.13i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0782754 + 1.13985i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0782754 + 1.13985i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.25 - 4.62i)T \) |
| 5 | \( 1 + (51.3 + 21.9i)T \) |
good | 3 | \( 1 + 1.29T + 243T^{2} \) |
| 7 | \( 1 - 170. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 39.8iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 537.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.35e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.03e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.61e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.91e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 9.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.29e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.21e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.93e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.89e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.89e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.23e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 4.88e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 882.T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.81e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.43e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 4.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.58e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.67e4iT - 8.58e9T^{2} \) |
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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
−15.51710426898917526744989440754, −14.89388929034968764078270123311, −13.39172977071340185557618198790, −12.19481426796955848466757806014, −11.39811904243063739376943910417, −8.788970013452624653688526125717, −8.260200957242783895807526977916, −6.33528862834168971237106335048, −5.10068830901643426837830582629, −3.29020830296107357514029930785,
0.54161663991036587798235947678, 3.17671049346862884973479420855, 4.48259969964408069295435128585, 6.46643541267537917551314523457, 8.195689399627203890436463479205, 10.07358223969028083824596944115, 11.15985648678301127157693059355, 11.89862380621203462896492517680, 13.53107280115793823366085160697, 14.24640713146177594601730572879