| L(s) = 1 | + (−0.686 − 0.727i)2-s + (0.224 − 1.71i)3-s + (−0.0581 + 0.998i)4-s + (2.24 − 0.262i)5-s + (−1.40 + 1.01i)6-s + (3.76 + 1.88i)7-s + (0.766 − 0.642i)8-s + (−2.89 − 0.771i)9-s + (−1.73 − 1.45i)10-s + (−1.42 − 3.29i)11-s + (1.70 + 0.323i)12-s + (−4.26 − 1.01i)13-s + (−1.20 − 4.03i)14-s + (0.0533 − 3.91i)15-s + (−0.993 − 0.116i)16-s + (−0.241 + 1.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.485 − 0.514i)2-s + (0.129 − 0.991i)3-s + (−0.0290 + 0.499i)4-s + (1.00 − 0.117i)5-s + (−0.572 + 0.414i)6-s + (1.42 + 0.714i)7-s + (0.270 − 0.227i)8-s + (−0.966 − 0.257i)9-s + (−0.547 − 0.459i)10-s + (−0.428 − 0.994i)11-s + (0.491 + 0.0935i)12-s + (−1.18 − 0.280i)13-s + (−0.322 − 1.07i)14-s + (0.0137 − 1.01i)15-s + (−0.248 − 0.0290i)16-s + (−0.0585 + 0.331i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.860977 - 0.705693i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.860977 - 0.705693i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.686 + 0.727i)T \) |
| 3 | \( 1 + (-0.224 + 1.71i)T \) |
| good | 5 | \( 1 + (-2.24 + 0.262i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (-3.76 - 1.88i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (1.42 + 3.29i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (4.26 + 1.01i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (0.241 - 1.36i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.883 - 5.00i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (0.261 - 0.131i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (-2.13 + 7.13i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (-6.75 - 4.44i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (2.90 - 1.05i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (5.07 - 5.37i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-1.66 + 2.23i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (0.834 - 0.548i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (5.96 - 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.66 - 8.49i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (0.863 + 14.8i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (1.60 + 5.34i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (-4.88 - 4.10i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-4.66 + 3.91i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-8.31 - 8.81i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (-2.12 - 2.25i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (0.706 - 0.592i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.85 - 0.216i)T + (94.3 + 22.3i)T^{2} \) |
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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
−12.39283013007837903043488158364, −11.82983516981398188580932045694, −10.71343076052354913184974299819, −9.564439437432719202855688696026, −8.303972612697615050120082459864, −7.909094436101060101991462376049, −6.17334945716835386011210924223, −5.15953566076887697480928732743, −2.69040187217130900566003472437, −1.60970847142071363158121886695,
2.23267783575360868593342997522, 4.68822441305766771718200806987, 5.15234379411811509904703396456, 6.91228500039942436959242725255, 7.952183288933059962937843141307, 9.189930189610492671919700858000, 9.997807513571883671086422882352, 10.67625613763018563661901398818, 11.78204588971976333806584346339, 13.52951646849528027988487835532
