| L(s) = 1 | + (1.23 + 3.80i)2-s + (−6.59 + 20.2i)3-s + (−12.9 + 9.40i)4-s − 48.6·5-s − 85.3·6-s + (4.11 − 2.98i)7-s + (−51.7 − 37.6i)8-s + (−171. − 124. i)9-s + (−60.1 − 185. i)10-s + (122. − 88.7i)11-s + (−105. − 324. i)12-s + (−45.6 + 140. i)13-s + (16.4 + 11.9i)14-s + (320. − 987. i)15-s + (79.1 − 243. i)16-s + (654. + 475. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.422 + 1.30i)3-s + (−0.404 + 0.293i)4-s − 0.870·5-s − 0.967·6-s + (0.0317 − 0.0230i)7-s + (−0.286 − 0.207i)8-s + (−0.705 − 0.512i)9-s + (−0.190 − 0.585i)10-s + (0.304 − 0.221i)11-s + (−0.211 − 0.650i)12-s + (−0.0749 + 0.230i)13-s + (0.0224 + 0.0163i)14-s + (0.368 − 1.13i)15-s + (0.0772 − 0.237i)16-s + (0.549 + 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.266866 - 0.324221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.266866 - 0.324221i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 - 3.80i)T \) |
| 31 | \( 1 + (4.76e3 - 2.43e3i)T \) |
| good | 3 | \( 1 + (6.59 - 20.2i)T + (-196. - 142. i)T^{2} \) |
| 5 | \( 1 + 48.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + (-4.11 + 2.98i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 11 | \( 1 + (-122. + 88.7i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (45.6 - 140. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-654. - 475. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (211. + 649. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (3.23e3 + 2.34e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (361. + 1.11e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 37 | \( 1 + 2.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.22e3 + 3.78e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + (-186. - 575. i)T + (-1.18e8 + 8.64e7i)T^{2} \) |
| 47 | \( 1 + (7.18e3 - 2.21e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.70e4 - 1.23e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.49e3 + 4.59e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + 1.29e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.76e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-1.37e4 - 9.99e3i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-2.60e4 + 1.89e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (8.09e4 + 5.88e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-906. - 2.79e3i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-1.05e4 + 7.67e3i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (4.04e4 - 2.94e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−15.05366983203451360169808231838, −14.12795234226408336562797325169, −12.45977381399142544640895555703, −11.36076328013341509969593594125, −10.22577540105409266580236783169, −8.964474289462979037800904804867, −7.67453774494397541883405069418, −6.03146128926962257842217943645, −4.61602466675359806943380533649, −3.70793544851084157688544121367,
0.19397267290693290543879634023, 1.76246170833392479257160253158, 3.75953163175971192319930270471, 5.64029477935234889446865174717, 7.15059170083709125244222334607, 8.163877574394754446790139789231, 9.909491323697345234246620612029, 11.50327954288430393288834129098, 11.98557167355584397558012962062, 12.91847808335624894118886397457
