| L(s) = 1 | + (−8.68 + 8.05i)2-s + (−21.8 + 3.28i)3-s + (−27.7 + 370. i)4-s + (−773. + 1.97e3i)5-s + (162. − 204. i)6-s + (−2.44e3 − 5.86e3i)7-s + (−6.52e3 − 8.18e3i)8-s + (−1.83e4 + 5.65e3i)9-s + (−9.16e3 − 2.33e4i)10-s + (−3.53e4 − 1.09e4i)11-s + (−612. − 8.17e3i)12-s + (3.08e4 + 1.34e5i)13-s + (6.85e4 + 3.11e4i)14-s + (1.03e4 − 4.55e4i)15-s + (−6.54e4 − 9.87e3i)16-s + (1.43e5 − 9.76e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.383 + 0.356i)2-s + (−0.155 + 0.0234i)3-s + (−0.0542 + 0.723i)4-s + (−0.553 + 1.41i)5-s + (0.0513 − 0.0643i)6-s + (−0.385 − 0.922i)7-s + (−0.563 − 0.706i)8-s + (−0.931 + 0.287i)9-s + (−0.289 − 0.738i)10-s + (−0.728 − 0.224i)11-s + (−0.00852 − 0.113i)12-s + (0.299 + 1.31i)13-s + (0.476 + 0.216i)14-s + (0.0530 − 0.232i)15-s + (−0.249 − 0.0376i)16-s + (0.415 − 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0588482 - 0.0306834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0588482 - 0.0306834i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.44e3 + 5.86e3i)T \) |
| good | 2 | \( 1 + (8.68 - 8.05i)T + (38.2 - 510. i)T^{2} \) |
| 3 | \( 1 + (21.8 - 3.28i)T + (1.88e4 - 5.80e3i)T^{2} \) |
| 5 | \( 1 + (773. - 1.97e3i)T + (-1.43e6 - 1.32e6i)T^{2} \) |
| 11 | \( 1 + (3.53e4 + 1.09e4i)T + (1.94e9 + 1.32e9i)T^{2} \) |
| 13 | \( 1 + (-3.08e4 - 1.34e5i)T + (-9.55e9 + 4.60e9i)T^{2} \) |
| 17 | \( 1 + (-1.43e5 + 9.76e4i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (-4.08e5 - 7.08e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (1.28e6 + 8.78e5i)T + (6.58e11 + 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-3.86e6 - 1.86e6i)T + (9.04e12 + 1.13e13i)T^{2} \) |
| 31 | \( 1 + (-1.00e6 + 1.73e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (1.29e6 + 1.72e7i)T + (-1.28e14 + 1.93e13i)T^{2} \) |
| 41 | \( 1 + (-9.71e6 - 1.21e7i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 43 | \( 1 + (2.59e7 - 3.24e7i)T + (-1.11e14 - 4.89e14i)T^{2} \) |
| 47 | \( 1 + (-3.60e7 + 3.34e7i)T + (8.36e13 - 1.11e15i)T^{2} \) |
| 53 | \( 1 + (-3.71e6 + 4.96e7i)T + (-3.26e15 - 4.91e14i)T^{2} \) |
| 59 | \( 1 + (-2.99e7 - 7.63e7i)T + (-6.35e15 + 5.89e15i)T^{2} \) |
| 61 | \( 1 + (7.95e6 + 1.06e8i)T + (-1.15e16 + 1.74e15i)T^{2} \) |
| 67 | \( 1 + (4.31e7 - 7.46e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-8.62e6 + 4.15e6i)T + (2.85e16 - 3.58e16i)T^{2} \) |
| 73 | \( 1 + (2.29e8 + 2.13e8i)T + (4.39e15 + 5.87e16i)T^{2} \) |
| 79 | \( 1 + (-5.72e7 - 9.91e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (1.56e7 - 6.84e7i)T + (-1.68e17 - 8.11e16i)T^{2} \) |
| 89 | \( 1 + (5.04e8 - 1.55e8i)T + (2.89e17 - 1.97e17i)T^{2} \) |
| 97 | \( 1 + 1.59e9T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.75711374337895648145515436711, −12.07850824761857065729472346291, −11.10084569851513692935999645394, −9.973315677929942440666745306299, −8.234786608552811952248907203487, −7.32961288073216075855983594333, −6.29712806454976899340678568863, −3.90552736893347192200055732450, −2.86781533653369372159643612077, −0.03309931579322917691992287674,
0.930430553658415735687013561142, 2.81863940408267374739092764161, 5.10034137461088733459108620156, 5.79513973034523565792495550935, 8.204125945383910725823151605426, 8.954599906824759532666057178188, 10.15887775279545973530539276223, 11.63224197596510195782346809518, 12.37499846098713260152523486171, 13.64955056671518551604823102130
