| L(s) = 1 | + (−1.36 − 0.380i)2-s + (1.70 − 0.284i)3-s + (0.0120 + 0.00727i)4-s + (−2.11 + 0.0820i)5-s + (−2.44 − 0.261i)6-s + (1.64 + 0.257i)7-s + (1.93 + 2.05i)8-s + (2.83 − 0.971i)9-s + (2.92 + 0.692i)10-s + (−0.699 − 5.12i)11-s + (0.0226 + 0.00899i)12-s + (3.44 − 5.02i)13-s + (−2.15 − 0.979i)14-s + (−3.58 + 0.741i)15-s + (−1.87 − 3.56i)16-s + (−1.16 + 0.584i)17-s + ⋯ |
| L(s) = 1 | + (−0.966 − 0.269i)2-s + (0.986 − 0.164i)3-s + (0.00602 + 0.00363i)4-s + (−0.945 + 0.0366i)5-s + (−0.997 − 0.106i)6-s + (0.623 + 0.0974i)7-s + (0.683 + 0.724i)8-s + (0.946 − 0.323i)9-s + (0.924 + 0.219i)10-s + (−0.210 − 1.54i)11-s + (0.00653 + 0.00259i)12-s + (0.954 − 1.39i)13-s + (−0.576 − 0.261i)14-s + (−0.926 + 0.191i)15-s + (−0.469 − 0.890i)16-s + (−0.282 + 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.711696 - 0.542260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.711696 - 0.542260i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.70 + 0.284i)T \) |
| good | 2 | \( 1 + (1.36 + 0.380i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (2.11 - 0.0820i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-1.64 - 0.257i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.699 + 5.12i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (-3.44 + 5.02i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (1.16 - 0.584i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.0591 - 1.01i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (0.0433 - 0.112i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (2.48 + 1.77i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (-2.39 + 2.74i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-4.61 - 6.20i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (0.596 - 2.31i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-3.38 - 3.07i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-7.20 - 8.25i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (0.302 - 1.71i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-8.73 + 3.56i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (-1.96 + 1.18i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (9.88 - 7.06i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-1.31 - 4.38i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (4.13 - 0.979i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (10.9 - 10.7i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (3.80 + 14.7i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (3.04 - 10.1i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (11.8 + 0.461i)T + (96.7 + 7.51i)T^{2} \) |
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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
−11.49141617898419921259967174871, −10.92177007945167338148066740017, −9.887693604366256730175894584607, −8.637536405653337145368398309948, −8.213276280868385386753296735429, −7.68666483928655693842376203514, −5.81724125219116982574927075811, −4.18527919703334796990538630145, −2.93712518072897562817672850772, −1.01745512570601493935931686471,
1.83023669900926349935102953253, 3.96902947257221708834181352032, 4.51297629800983172997030104142, 7.05090579243458757482649107858, 7.50129781678702836811655042264, 8.522161719727996390824412586256, 9.111198375624591891776353510526, 10.07522785943922450758422883547, 11.14300135470232778005879087189, 12.28536088131161967760431180687
