| L(s) = 1 | + (−2.05 − 0.0266i)2-s + (0.611 + 1.62i)3-s + (2.23 + 0.0578i)4-s + (−2.65 − 0.814i)5-s + (−1.21 − 3.35i)6-s + (1.70 − 4.26i)7-s + (−0.488 − 0.0189i)8-s + (−2.25 + 1.98i)9-s + (5.44 + 1.74i)10-s + (−0.0568 − 0.0323i)11-s + (1.27 + 3.65i)12-s + (−0.493 − 1.10i)13-s + (−3.63 + 8.72i)14-s + (−0.306 − 4.80i)15-s + (−3.46 − 0.179i)16-s + (−0.381 − 0.173i)17-s + ⋯ |
| L(s) = 1 | + (−1.45 − 0.0188i)2-s + (0.353 + 0.935i)3-s + (1.11 + 0.0289i)4-s + (−1.18 − 0.364i)5-s + (−0.496 − 1.36i)6-s + (0.646 − 1.61i)7-s + (−0.172 − 0.00670i)8-s + (−0.750 + 0.660i)9-s + (1.72 + 0.552i)10-s + (−0.0171 − 0.00974i)11-s + (0.367 + 1.05i)12-s + (−0.136 − 0.306i)13-s + (−0.970 + 2.33i)14-s + (−0.0790 − 1.24i)15-s + (−0.865 − 0.0448i)16-s + (−0.0924 − 0.0420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0181767 + 0.134448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0181767 + 0.134448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.611 - 1.62i)T \) |
| good | 2 | \( 1 + (2.05 + 0.0266i)T + (1.99 + 0.0517i)T^{2} \) |
| 5 | \( 1 + (2.65 + 0.814i)T + (4.14 + 2.80i)T^{2} \) |
| 7 | \( 1 + (-1.70 + 4.26i)T + (-5.06 - 4.83i)T^{2} \) |
| 11 | \( 1 + (0.0568 + 0.0323i)T + (5.62 + 9.45i)T^{2} \) |
| 13 | \( 1 + (0.493 + 1.10i)T + (-8.67 + 9.68i)T^{2} \) |
| 17 | \( 1 + (0.381 + 0.173i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.205 - 0.300i)T + (-6.84 + 17.7i)T^{2} \) |
| 23 | \( 1 + (2.62 - 8.02i)T + (-18.5 - 13.6i)T^{2} \) |
| 29 | \( 1 + (2.39 - 1.53i)T + (12.1 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.59 - 9.54i)T + (-26.7 + 15.6i)T^{2} \) |
| 37 | \( 1 + (-0.472 - 0.0927i)T + (34.2 + 13.9i)T^{2} \) |
| 41 | \( 1 + (7.28 - 2.76i)T + (30.7 - 27.1i)T^{2} \) |
| 43 | \( 1 + (3.00 - 1.31i)T + (29.1 - 31.6i)T^{2} \) |
| 47 | \( 1 + (-3.40 + 12.5i)T + (-40.5 - 23.7i)T^{2} \) |
| 53 | \( 1 + (5.59 - 5.92i)T + (-3.08 - 52.9i)T^{2} \) |
| 59 | \( 1 + (-8.32 + 1.41i)T + (55.6 - 19.4i)T^{2} \) |
| 61 | \( 1 + (3.89 - 7.15i)T + (-33.1 - 51.1i)T^{2} \) |
| 67 | \( 1 + (0.604 - 0.313i)T + (38.6 - 54.7i)T^{2} \) |
| 71 | \( 1 + (7.25 + 5.62i)T + (17.7 + 68.7i)T^{2} \) |
| 73 | \( 1 + (11.1 - 3.56i)T + (59.3 - 42.4i)T^{2} \) |
| 79 | \( 1 + (0.844 - 4.45i)T + (-73.5 - 28.9i)T^{2} \) |
| 83 | \( 1 + (-0.152 - 0.935i)T + (-78.7 + 26.3i)T^{2} \) |
| 89 | \( 1 + (11.5 + 4.71i)T + (63.5 + 62.3i)T^{2} \) |
| 97 | \( 1 + (-0.977 + 4.24i)T + (-87.2 - 42.4i)T^{2} \) |
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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
−10.35624558222441710611880983308, −10.13356302739954478022113560964, −8.967928832105491648310499821672, −8.265198848244707703982447241291, −7.67931184779042602606210813271, −7.08225026110229006602585923838, −5.10187212955705183340494638939, −4.24178192000790612124045969819, −3.42896509263639477955543693812, −1.40104828883233665671432543179,
0.11379843652911474774395814429, 1.88639803594724137132995966458, 2.73694362854808438329362578676, 4.39590123506854069934681134517, 5.94836739689170216218499276495, 6.88540530693206325535736966726, 7.83094950757769049607721941517, 8.235430266768104105050993709174, 8.838882889743076306653896316939, 9.685986037846209286599899032417
