| L(s) = 1 | + (−1.44 + 1.72i)2-s + (0.130 − 1.72i)3-s + (−0.529 − 3.00i)4-s + (−1.80 + 1.31i)5-s + (2.78 + 2.71i)6-s + (−3.05 − 0.538i)7-s + (2.03 + 1.17i)8-s + (−2.96 − 0.451i)9-s + (0.340 − 5.01i)10-s + (−3.82 + 1.39i)11-s + (−5.25 + 0.522i)12-s + (−0.913 − 1.08i)13-s + (5.33 − 4.47i)14-s + (2.04 + 3.29i)15-s + (0.757 − 0.275i)16-s + (5.52 − 3.18i)17-s + ⋯ |
| L(s) = 1 | + (−1.02 + 1.21i)2-s + (0.0754 − 0.997i)3-s + (−0.264 − 1.50i)4-s + (−0.807 + 0.589i)5-s + (1.13 + 1.11i)6-s + (−1.15 − 0.203i)7-s + (0.720 + 0.416i)8-s + (−0.988 − 0.150i)9-s + (0.107 − 1.58i)10-s + (−1.15 + 0.419i)11-s + (−1.51 + 0.150i)12-s + (−0.253 − 0.301i)13-s + (1.42 − 1.19i)14-s + (0.526 + 0.849i)15-s + (0.189 − 0.0689i)16-s + (1.33 − 0.773i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0161044 - 0.0346880i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0161044 - 0.0346880i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.130 + 1.72i)T \) |
| 5 | \( 1 + (1.80 - 1.31i)T \) |
| good | 2 | \( 1 + (1.44 - 1.72i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (3.05 + 0.538i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (3.82 - 1.39i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.913 + 1.08i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.52 + 3.18i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.59 - 2.76i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.75 - 0.838i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.87 + 2.41i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.453 - 2.57i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (0.545 - 0.314i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.50 + 3.78i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.01 - 8.29i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (8.23 + 1.45i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 0.0211iT - 53T^{2} \) |
| 59 | \( 1 + (-1.27 - 0.463i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.492 + 2.79i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.983 - 1.17i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (7.47 + 12.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.31 + 3.64i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.125 + 0.105i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.27 + 2.70i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (7.60 - 13.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.59 + 12.6i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−12.85480918695506026019258418504, −12.02221215554000669687829221559, −10.42656297935225913692537753087, −9.574284614400268201488848674345, −7.991169576845835398610864281130, −7.66328944285511328321902398700, −6.71005079507135616203866392464, −5.69997760651640912428512634261, −3.10228243809699700338585923883, −0.05059632911630849589756276096,
2.87022958998112484849439838815, 3.90991907102687407536819453938, 5.63035923449498478549733395879, 7.88133624443188770038704153269, 8.716298593907268227250136204192, 9.670976316198739816786255845214, 10.36611415625652043700458384635, 11.31978071841546402272159360418, 12.29660706222166341732851529419, 13.07203354418222152911907296767
