| L(s) = 1 | + (−0.973 − 0.230i)2-s + (0.877 + 1.49i)3-s + (0.893 + 0.448i)4-s + (2.62 − 3.52i)5-s + (−0.509 − 1.65i)6-s + (−1.50 − 0.990i)7-s + (−0.766 − 0.642i)8-s + (−1.45 + 2.62i)9-s + (−3.36 + 2.82i)10-s + (3.24 − 0.378i)11-s + (0.114 + 1.72i)12-s + (0.00477 − 0.0159i)13-s + (1.23 + 1.31i)14-s + (7.55 + 0.822i)15-s + (0.597 + 0.802i)16-s + (−0.143 − 0.812i)17-s + ⋯ |
| L(s) = 1 | + (−0.688 − 0.163i)2-s + (0.506 + 0.862i)3-s + (0.446 + 0.224i)4-s + (1.17 − 1.57i)5-s + (−0.208 − 0.675i)6-s + (−0.568 − 0.374i)7-s + (−0.270 − 0.227i)8-s + (−0.486 + 0.873i)9-s + (−1.06 + 0.892i)10-s + (0.977 − 0.114i)11-s + (0.0330 + 0.498i)12-s + (0.00132 − 0.00442i)13-s + (0.330 + 0.350i)14-s + (1.95 + 0.212i)15-s + (0.149 + 0.200i)16-s + (−0.0347 − 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10181 - 0.0895495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10181 - 0.0895495i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.973 + 0.230i)T \) |
| 3 | \( 1 + (-0.877 - 1.49i)T \) |
| good | 5 | \( 1 + (-2.62 + 3.52i)T + (-1.43 - 4.78i)T^{2} \) |
| 7 | \( 1 + (1.50 + 0.990i)T + (2.77 + 6.42i)T^{2} \) |
| 11 | \( 1 + (-3.24 + 0.378i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (-0.00477 + 0.0159i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (0.143 + 0.812i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (1.34 - 7.64i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (-0.461 + 0.303i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (2.21 - 2.35i)T + (-1.68 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.266 - 4.57i)T + (-30.7 + 3.59i)T^{2} \) |
| 37 | \( 1 + (-3.29 - 1.20i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (10.4 - 2.48i)T + (36.6 - 18.4i)T^{2} \) |
| 43 | \( 1 + (2.30 - 5.34i)T + (-29.5 - 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.636 + 10.9i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (2.34 + 4.06i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.16 + 0.369i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (-1.76 + 0.884i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (1.44 + 1.53i)T + (-3.89 + 66.8i)T^{2} \) |
| 71 | \( 1 + (7.37 - 6.18i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-4.83 - 4.05i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.63 - 1.09i)T + (70.5 + 35.4i)T^{2} \) |
| 83 | \( 1 + (-6.63 - 1.57i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (-9.83 - 8.25i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (5.09 + 6.84i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−12.87686647908916783079799081887, −11.83437190152924576890827725298, −10.24180932550723432911612006411, −9.779089678629207607375163330743, −8.923227660648326370586686053665, −8.239404388659895987120349134717, −6.35669066024381414432637175117, −5.10542052736703458464545806286, −3.70164752861829993639222543237, −1.67814170164546701544060441049,
2.06121770578590649417626780896, 3.09744677970145342753991119256, 6.07115391858262643045071523791, 6.59976583628429873348661352063, 7.43434994849818371929110716373, 9.065686717792946236590150560129, 9.556162731712846603003194719133, 10.79365175011425724904878708393, 11.73514548871827720100378120386, 13.14269558038475120596119697095
