| 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
−13.14269558038475120596119697095, −11.73514548871827720100378120386, −10.79365175011425724904878708393, −9.556162731712846603003194719133, −9.065686717792946236590150560129, −7.43434994849818371929110716373, −6.59976583628429873348661352063, −6.07115391858262643045071523791, −3.09744677970145342753991119256, −2.06121770578590649417626780896,
1.67814170164546701544060441049, 3.70164752861829993639222543237, 5.10542052736703458464545806286, 6.35669066024381414432637175117, 8.239404388659895987120349134717, 8.923227660648326370586686053665, 9.779089678629207607375163330743, 10.24180932550723432911612006411, 11.83437190152924576890827725298, 12.87686647908916783079799081887
