| L(s) = 1 | + (0.0300 − 0.170i)2-s + (1.85 + 0.673i)4-s + (−2.86 + 2.40i)5-s + (−2.84 + 1.03i)7-s + (0.343 − 0.594i)8-s + (0.323 + 0.559i)10-s + (−1.90 − 1.60i)11-s + (−0.132 − 0.753i)13-s + (0.0910 + 0.516i)14-s + (2.92 + 2.45i)16-s + (−2.31 − 4.00i)17-s + (0.305 − 0.529i)19-s + (−6.91 + 2.51i)20-s + (−0.330 + 0.277i)22-s + (−6.13 − 2.23i)23-s + ⋯ |
| L(s) = 1 | + (0.0212 − 0.120i)2-s + (0.925 + 0.336i)4-s + (−1.28 + 1.07i)5-s + (−1.07 + 0.391i)7-s + (0.121 − 0.210i)8-s + (0.102 + 0.177i)10-s + (−0.575 − 0.482i)11-s + (−0.0368 − 0.208i)13-s + (0.0243 + 0.138i)14-s + (0.731 + 0.614i)16-s + (−0.560 − 0.970i)17-s + (0.0701 − 0.121i)19-s + (−1.54 + 0.562i)20-s + (−0.0703 + 0.0590i)22-s + (−1.27 − 0.465i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0151093 - 0.259417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0151093 - 0.259417i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.0300 + 0.170i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (2.86 - 2.40i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (2.84 - 1.03i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (1.90 + 1.60i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.132 + 0.753i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (2.31 + 4.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.305 + 0.529i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.13 + 2.23i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.13 - 6.45i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (6.15 + 2.24i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (2.47 + 4.29i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.913 - 5.18i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.26 - 3.58i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (1.04 - 0.378i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 8.84T + 53T^{2} \) |
| 59 | \( 1 + (9.07 - 7.61i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (7.69 - 2.80i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.210 + 1.19i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.45 - 4.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.14 - 3.72i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.04 - 11.6i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.56 + 8.87i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (3.76 - 6.52i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.726 + 0.609i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.92419486764217115274064395761, −10.30365484047945026432745356885, −9.112604707071176053305445814511, −7.977122115163749739467649107801, −7.31514557107655790282168384618, −6.65715841420751562261648053458, −5.75269798916669007351052332490, −4.04424404555333537961230938827, −3.13422542463803525097879346158, −2.57359717610009274468463127718,
0.12012016681893575082245411132, 1.88913286529967702402153895365, 3.48402520877320597201429315941, 4.30031986562471751465470205455, 5.51397813814039058029563747474, 6.47535412340687073546986428867, 7.45034794592814035355060386283, 7.990426988817516971589820277457, 9.070642041370740357088405122943, 10.04865187376510803649149408683
