| L(s) = 1 | + (−1.25 + 0.659i)2-s + (0.590 + 1.62i)3-s + (1.13 − 1.64i)4-s + (−1.35 + 1.28i)5-s + (−1.81 − 1.64i)6-s + (1.85 − 0.217i)7-s + (−0.327 + 2.80i)8-s + (−2.30 + 1.92i)9-s + (0.854 − 2.49i)10-s + (3.80 − 1.13i)11-s + (3.35 + 0.868i)12-s + (6.21 − 0.362i)13-s + (−2.18 + 1.49i)14-s + (−2.88 − 1.45i)15-s + (−1.44 − 3.73i)16-s + (0.526 + 0.191i)17-s + ⋯ |
| L(s) = 1 | + (−0.884 + 0.466i)2-s + (0.340 + 0.940i)3-s + (0.565 − 0.824i)4-s + (−0.607 + 0.572i)5-s + (−0.739 − 0.672i)6-s + (0.702 − 0.0821i)7-s + (−0.115 + 0.993i)8-s + (−0.767 + 0.640i)9-s + (0.270 − 0.789i)10-s + (1.14 − 0.343i)11-s + (0.968 + 0.250i)12-s + (1.72 − 0.100i)13-s + (−0.583 + 0.400i)14-s + (−0.745 − 0.375i)15-s + (−0.360 − 0.932i)16-s + (0.127 + 0.0464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.585389 + 0.985537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.585389 + 0.985537i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.25 - 0.659i)T \) |
| 3 | \( 1 + (-0.590 - 1.62i)T \) |
| good | 5 | \( 1 + (1.35 - 1.28i)T + (0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-1.85 + 0.217i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (-3.80 + 1.13i)T + (9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (-6.21 + 0.362i)T + (12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (-0.526 - 0.191i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-1.89 - 5.21i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (7.31 + 0.854i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (-3.07 - 6.13i)T + (-17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (-4.15 + 5.58i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (5.63 + 6.71i)T + (-6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (0.461 - 0.303i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (1.38 - 5.82i)T + (-38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-1.50 - 2.01i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (-2.82 + 1.63i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.31 - 1.29i)T + (49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (8.13 + 3.50i)T + (41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (6.29 - 12.5i)T + (-40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (1.76 + 9.98i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.882 + 5.00i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-8.60 - 5.66i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (8.77 - 13.3i)T + (-32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (-1.80 + 10.2i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.27 - 2.41i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.74094062777393289023627889988, −9.955766885648515069500047558910, −8.960924477887137912393185268043, −8.299439847458656913298250497160, −7.69711106771742438204823325132, −6.37584149102851126628822083482, −5.64740912567169954256969045408, −4.16975043437766390599889988370, −3.36034352577592466601350526948, −1.54603912801523898028102096345,
0.918081035799662591137338961585, 1.84782704816130892797895379685, 3.36795504027856494685127987169, 4.37359383643817132822269762443, 6.17063204716315097680176761562, 6.92916287165819335452731060971, 8.005155442343591305992261782778, 8.503488398990725306940156730243, 9.043690428928683044441482583958, 10.23734134508810165809062656935
