| L(s) = 1 | + (−3.12 − 4.30i)2-s + (13.7 + 42.4i)3-s + (11.0 − 33.9i)4-s + (141. + 102. i)5-s + (139. − 191. i)6-s + (43.6 + 14.1i)7-s + (−504. + 163. i)8-s + (−1.01e3 + 740. i)9-s − 928. i·10-s + (288. − 1.29e3i)11-s + 1.59e3·12-s + (−610. − 840. i)13-s + (−75.4 − 232. i)14-s + (−2.40e3 + 7.40e3i)15-s + (435. + 316. i)16-s + (4.59e3 − 6.31e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.390 − 0.538i)2-s + (0.510 + 1.57i)3-s + (0.172 − 0.530i)4-s + (1.13 + 0.821i)5-s + (0.645 − 0.888i)6-s + (0.127 + 0.0413i)7-s + (−0.985 + 0.320i)8-s + (−1.39 + 1.01i)9-s − 0.928i·10-s + (0.217 − 0.976i)11-s + 0.920·12-s + (−0.277 − 0.382i)13-s + (−0.0274 − 0.0846i)14-s + (−0.712 + 2.19i)15-s + (0.106 + 0.0772i)16-s + (0.934 − 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.38976 + 0.339392i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38976 + 0.339392i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-288. + 1.29e3i)T \) |
| good | 2 | \( 1 + (3.12 + 4.30i)T + (-19.7 + 60.8i)T^{2} \) |
| 3 | \( 1 + (-13.7 - 42.4i)T + (-589. + 428. i)T^{2} \) |
| 5 | \( 1 + (-141. - 102. i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (-43.6 - 14.1i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (610. + 840. i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (-4.59e3 + 6.31e3i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (6.66e3 - 2.16e3i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 - 3.54e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.08e4 + 3.51e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (1.70e4 - 1.24e4i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (-1.19e4 + 3.68e4i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (-7.36e4 + 2.39e4i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 - 9.04e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (7.06e3 + 2.17e4i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (-2.98e4 + 2.17e4i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (9.31e4 - 2.86e5i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (1.82e5 - 2.51e5i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 - 2.03e4T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-1.16e5 - 8.47e4i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (2.98e5 + 9.70e4i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-2.88e5 - 3.97e5i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (1.97e5 - 2.71e5i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 - 4.84e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-3.11e5 + 2.25e5i)T + (2.57e11 - 7.92e11i)T^{2} \) |
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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
−19.51261863630320115949482842789, −18.17079462421371679760678171568, −16.44602836536978150317176248030, −14.86549189730017244028611914914, −14.15459704865458465555667747173, −11.05531828132662548477522581451, −10.13504396535589638984929042611, −9.166637775943503917945428415735, −5.66532087746546961764815087075, −2.85316763931066862683951758559,
1.85372356247436253751613081470, 6.37154437188186117799684448707, 7.83187151481982582705989319230, 9.179157085533544197139749995553, 12.40127304163340737333389682252, 13.03842693145228509416808416744, 14.66675172122660554855324152404, 17.02246542138489804664502762843, 17.49774954733297533628261731516, 18.84770255736696575319394911347
