| L(s) = 1 | + (0.592 + 0.430i)2-s + (1.83 − 5.63i)3-s + (−2.30 − 7.09i)4-s + (10.4 − 7.55i)5-s + (3.51 − 2.55i)6-s + (5.23 + 16.0i)7-s + (3.49 − 10.7i)8-s + (−6.58 − 4.78i)9-s + 9.41·10-s − 44.2·12-s + (−60.3 − 43.8i)13-s + (−3.82 + 11.7i)14-s + (−23.5 − 72.4i)15-s + (−41.6 + 30.2i)16-s + (66.9 − 48.6i)17-s + (−1.84 − 5.66i)18-s + ⋯ |
| L(s) = 1 | + (0.209 + 0.152i)2-s + (0.352 − 1.08i)3-s + (−0.288 − 0.887i)4-s + (0.930 − 0.675i)5-s + (0.238 − 0.173i)6-s + (0.282 + 0.869i)7-s + (0.154 − 0.475i)8-s + (−0.244 − 0.177i)9-s + 0.297·10-s − 1.06·12-s + (−1.28 − 0.936i)13-s + (−0.0731 + 0.224i)14-s + (−0.405 − 1.24i)15-s + (−0.650 + 0.472i)16-s + (0.955 − 0.694i)17-s + (−0.0241 − 0.0742i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.32987 - 1.66460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32987 - 1.66460i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.592 - 0.430i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (-1.83 + 5.63i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (-10.4 + 7.55i)T + (38.6 - 118. i)T^{2} \) |
| 7 | \( 1 + (-5.23 - 16.0i)T + (-277. + 201. i)T^{2} \) |
| 13 | \( 1 + (60.3 + 43.8i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-66.9 + 48.6i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (20.9 - 64.5i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 13.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-52.2 - 160. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-52.9 - 38.4i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-12.6 - 38.8i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-84.9 + 261. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 2.28T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-22.2 + 68.3i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-120. - 87.5i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-168. - 518. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (81.9 - 59.5i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 - 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-380. + 276. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-188. - 580. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-791. - 574. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (21.1 - 15.3i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 352.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (685. + 498. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−12.71161818321182146576115517292, −12.21438983944428892680529790234, −10.33100421279199554646190377592, −9.487516158784775366853733817696, −8.372095822359339257371595801428, −7.11320882630493183166468093526, −5.68639523871673855980070030525, −5.10524557464090069440185987082, −2.36614030496348924347731834874, −1.12867717999534965530719790410,
2.54100542514602026683780887478, 3.91440242997707404642592776207, 4.83360414669785642417351681811, 6.72874087294531829015610068451, 7.929088101588973237555439035932, 9.382237648239006936012511313487, 10.00022014592336457447120926603, 11.02589219743709818179509366459, 12.28271681331038343736894330203, 13.51780838384768065402071178380
