| L(s) = 1 | + (−0.393 + 1.01i)2-s + (−1.23 − 1.21i)3-s + (0.596 + 0.541i)4-s + (−1.58 − 1.13i)5-s + (1.72 − 0.777i)6-s + (−1.04 − 0.810i)7-s + (−2.74 + 1.37i)8-s + (0.0364 + 2.99i)9-s + (1.78 − 1.17i)10-s + (−2.98 + 4.35i)11-s + (−0.0760 − 1.39i)12-s + (−3.16 + 0.622i)13-s + (1.23 − 0.746i)14-s + (0.575 + 3.33i)15-s + (−0.168 − 1.73i)16-s + (−1.42 − 3.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.278 + 0.720i)2-s + (−0.711 − 0.702i)3-s + (0.298 + 0.270i)4-s + (−0.710 − 0.507i)5-s + (0.704 − 0.317i)6-s + (−0.395 − 0.306i)7-s + (−0.968 + 0.486i)8-s + (0.0121 + 0.999i)9-s + (0.563 − 0.370i)10-s + (−0.900 + 1.31i)11-s + (−0.0219 − 0.402i)12-s + (−0.878 + 0.172i)13-s + (0.330 − 0.199i)14-s + (0.148 + 0.860i)15-s + (−0.0421 − 0.432i)16-s + (−0.346 − 0.804i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00750159 - 0.163384i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00750159 - 0.163384i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.23 + 1.21i)T \) |
| good | 2 | \( 1 + (0.393 - 1.01i)T + (-1.48 - 1.34i)T^{2} \) |
| 5 | \( 1 + (1.58 + 1.13i)T + (1.61 + 4.73i)T^{2} \) |
| 7 | \( 1 + (1.04 + 0.810i)T + (1.74 + 6.77i)T^{2} \) |
| 11 | \( 1 + (2.98 - 4.35i)T + (-3.96 - 10.2i)T^{2} \) |
| 13 | \( 1 + (3.16 - 0.622i)T + (12.0 - 4.91i)T^{2} \) |
| 17 | \( 1 + (1.42 + 3.31i)T + (-11.6 + 12.3i)T^{2} \) |
| 19 | \( 1 + (-0.642 - 0.862i)T + (-5.44 + 18.2i)T^{2} \) |
| 23 | \( 1 + (1.77 + 0.723i)T + (16.4 + 16.1i)T^{2} \) |
| 29 | \( 1 + (5.83 - 3.21i)T + (15.4 - 24.5i)T^{2} \) |
| 31 | \( 1 + (-1.78 - 3.39i)T + (-17.5 + 25.5i)T^{2} \) |
| 37 | \( 1 + (-0.174 - 0.185i)T + (-2.15 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-4.05 + 5.02i)T + (-8.67 - 40.0i)T^{2} \) |
| 43 | \( 1 + (-5.39 - 5.28i)T + (0.833 + 42.9i)T^{2} \) |
| 47 | \( 1 + (0.388 - 0.736i)T + (-26.5 - 38.7i)T^{2} \) |
| 53 | \( 1 + (-7.49 - 2.72i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-3.47 + 0.269i)T + (58.2 - 9.11i)T^{2} \) |
| 61 | \( 1 + (7.83 - 7.10i)T + (5.90 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-0.871 - 0.481i)T + (35.7 + 56.6i)T^{2} \) |
| 71 | \( 1 + (-0.332 - 5.71i)T + (-70.5 + 8.24i)T^{2} \) |
| 73 | \( 1 + (6.58 + 4.33i)T + (28.9 + 67.0i)T^{2} \) |
| 79 | \( 1 + (8.62 - 1.34i)T + (75.2 - 24.1i)T^{2} \) |
| 83 | \( 1 + (-10.2 - 12.7i)T + (-17.5 + 81.1i)T^{2} \) |
| 89 | \( 1 + (-0.638 + 10.9i)T + (-88.3 - 10.3i)T^{2} \) |
| 97 | \( 1 + (15.4 - 11.0i)T + (31.4 - 91.7i)T^{2} \) |
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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
−12.31938379036365927653770721610, −12.02269742035699225530448697323, −10.82516379188119528658073668618, −9.627592596858168856780598315227, −8.275026814630918092665592410986, −7.30600838150222263777931137954, −7.09337444005996587353321427326, −5.61777301627550383374652079030, −4.52104678180035531517457444208, −2.44938992800475231874346226070,
0.14210895920515527588653706761, 2.75338784215527166294779536587, 3.81596347954468225636815334102, 5.52083080480152048045706538887, 6.28671664535435103777038681976, 7.66617926827544712479652957454, 9.080044366468800299337575713434, 10.00713788829523129949998517925, 10.83080550234987393306059339781, 11.33323462439335425548811671403
