| L(s) = 1 | + (−0.976 + 3.00i)2-s + (1.24 − 0.906i)3-s + (−1.60 − 1.16i)4-s + (−5.33 − 16.4i)5-s + (1.50 + 4.63i)6-s + (7.73 + 5.62i)7-s + (−15.3 + 11.1i)8-s + (−7.60 + 23.4i)9-s + 54.5·10-s + (−8.01 − 35.5i)11-s − 3.05·12-s + (−2.00 + 6.16i)13-s + (−24.4 + 17.7i)14-s + (−21.5 − 15.6i)15-s + (−23.4 − 72.2i)16-s + (29.2 + 89.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.345 + 1.06i)2-s + (0.239 − 0.174i)3-s + (−0.200 − 0.145i)4-s + (−0.477 − 1.46i)5-s + (0.102 + 0.315i)6-s + (0.417 + 0.303i)7-s + (−0.679 + 0.493i)8-s + (−0.281 + 0.867i)9-s + 1.72·10-s + (−0.219 − 0.975i)11-s − 0.0734·12-s + (−0.0427 + 0.131i)13-s + (−0.466 + 0.339i)14-s + (−0.370 − 0.269i)15-s + (−0.366 − 1.12i)16-s + (0.416 + 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.729693 + 0.350498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.729693 + 0.350498i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (8.01 + 35.5i)T \) |
| good | 2 | \( 1 + (0.976 - 3.00i)T + (-6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (-1.24 + 0.906i)T + (8.34 - 25.6i)T^{2} \) |
| 5 | \( 1 + (5.33 + 16.4i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (-7.73 - 5.62i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (2.00 - 6.16i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-29.2 - 89.9i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-42.5 + 30.9i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 - 82.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + (121. + 88.1i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-40.6 + 124. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-99.3 - 72.1i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (153. - 111. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 + 320.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-356. + 259. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (38.1 - 117. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-58.6 - 42.6i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-76.8 - 236. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + 128.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-306. - 944. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (418. + 303. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-177. + 546. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-58.1 - 178. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 42.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + (397. - 1.22e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
| 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
−20.28786627985796894958014324776, −18.90748982695659057791787391964, −17.03856452357373302680271326361, −16.42780093573001994995623095739, −15.09895292182853662893102926438, −13.30085885866207257890960675154, −11.61027965946623635218410291867, −8.697114451620767620632905733113, −7.957665078121613920800379562655, −5.42252837599943190984788832279,
3.15176628580912465142752062905, 7.11034028876100600297067250240, 9.659710248813991763362480086306, 10.93218691783569772035322059968, 12.06889652743494885961436630302, 14.44217615434636990452779235076, 15.41322427846005257782453366416, 17.91057116878277742061079180345, 18.69685110582044052912442412576, 20.10325599649215013599367418296
