| L(s) = 1 | + (14.3 − 10.4i)2-s + (−6.74 − 20.7i)3-s + (57.8 − 178. i)4-s + (53.3 + 38.7i)5-s + (−313. − 227. i)6-s + (−143. + 441. i)7-s + (−324. − 9.99e2i)8-s + (1.38e3 − 1.00e3i)9-s + 1.17e3·10-s + (3.30e3 + 2.92e3i)11-s − 4.08e3·12-s + (−1.06e4 + 7.72e3i)13-s + (2.54e3 + 7.83e3i)14-s + (444. − 1.36e3i)15-s + (4.28e3 + 3.11e3i)16-s + (−985. − 716. i)17-s + ⋯ |
| L(s) = 1 | + (1.26 − 0.922i)2-s + (−0.144 − 0.443i)3-s + (0.451 − 1.39i)4-s + (0.190 + 0.138i)5-s + (−0.592 − 0.430i)6-s + (−0.158 + 0.486i)7-s + (−0.224 − 0.690i)8-s + (0.632 − 0.459i)9-s + 0.370·10-s + (0.748 + 0.663i)11-s − 0.682·12-s + (−1.34 + 0.975i)13-s + (0.247 + 0.763i)14-s + (0.0340 − 0.104i)15-s + (0.261 + 0.190i)16-s + (−0.0486 − 0.0353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.93864 - 1.59601i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93864 - 1.59601i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-3.30e3 - 2.92e3i)T \) |
| good | 2 | \( 1 + (-14.3 + 10.4i)T + (39.5 - 121. i)T^{2} \) |
| 3 | \( 1 + (6.74 + 20.7i)T + (-1.76e3 + 1.28e3i)T^{2} \) |
| 5 | \( 1 + (-53.3 - 38.7i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (143. - 441. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (1.06e4 - 7.72e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (985. + 716. i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (1.14e4 + 3.53e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + 5.72e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (2.32e4 - 7.15e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-9.58e4 + 6.96e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (1.44e5 - 4.44e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (1.16e4 + 3.58e4i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 2.77e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (3.44e4 + 1.06e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-3.73e5 + 2.71e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-5.80e5 + 1.78e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.97e6 + 1.43e6i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 4.41e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (3.83e6 + 2.78e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (1.48e6 - 4.56e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-2.30e6 + 1.67e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-2.41e6 - 1.75e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 5.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-1.31e6 + 9.55e5i)T + (2.49e13 - 7.68e13i)T^{2} \) |
| 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
−19.06059469408481422692075048684, −17.43877947343821478521003257054, −15.21068044650213716639213592961, −13.98394804386498150730437362780, −12.50694383652468228248052580700, −11.79177102454518465526457844620, −9.760790625321949205197844376980, −6.62622131683166543477056061560, −4.46886525784018246800874937326, −2.13190909048913069312674524912,
4.02549322094841289518172379633, 5.65815485421936798523088124109, 7.50041785762810232358816758077, 10.14422940423149731808054605867, 12.38864892626618033836367022393, 13.70520391545713644173902129152, 14.91074954912829057725903801362, 16.20527985148465357452420459439, 17.13923835165602387140911610679, 19.41330683812908386114074889792
