| L(s) = 1 | + (−1.56 − 0.732i)3-s + (0.407 + 0.341i)5-s + (−0.507 − 0.184i)7-s + (1.92 + 2.29i)9-s + (1.49 − 1.25i)11-s + (0.696 − 3.94i)13-s + (−0.388 − 0.834i)15-s + (0.0114 − 0.0199i)17-s + (−1.25 − 2.17i)19-s + (0.661 + 0.661i)21-s + (6.43 − 2.34i)23-s + (−0.819 − 4.64i)25-s + (−1.33 − 5.02i)27-s + (−1.03 − 5.88i)29-s + (−3.81 + 1.38i)31-s + ⋯ |
| L(s) = 1 | + (−0.906 − 0.423i)3-s + (0.182 + 0.152i)5-s + (−0.191 − 0.0698i)7-s + (0.642 + 0.766i)9-s + (0.449 − 0.377i)11-s + (0.193 − 1.09i)13-s + (−0.100 − 0.215i)15-s + (0.00278 − 0.00483i)17-s + (−0.287 − 0.498i)19-s + (0.144 + 0.144i)21-s + (1.34 − 0.488i)23-s + (−0.163 − 0.929i)25-s + (−0.257 − 0.966i)27-s + (−0.192 − 1.09i)29-s + (−0.684 + 0.249i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.808466 - 0.584585i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.808466 - 0.584585i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.56 + 0.732i)T \) |
| good | 5 | \( 1 + (-0.407 - 0.341i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (0.507 + 0.184i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-1.49 + 1.25i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.696 + 3.94i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.0114 + 0.0199i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.25 + 2.17i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.43 + 2.34i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.03 + 5.88i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.81 - 1.38i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.58 + 6.21i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.30 - 7.42i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.23 + 3.55i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-10.4 - 3.81i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 2.91T + 53T^{2} \) |
| 59 | \( 1 + (-3.02 - 2.53i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (9.15 + 3.33i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.88 - 10.6i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.30 - 3.98i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.36 - 14.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.23 + 12.6i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.334 - 1.89i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.68 - 6.38i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.32 - 1.95i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.94462797363261135967894380498, −10.36702283001576579059589793315, −9.227886316023637314521202799199, −8.097056802047865341061097350301, −7.10103287926095881016010827160, −6.21383006832052271347838414068, −5.42895126463658669857494231204, −4.19072717321892399566436110607, −2.60768499841721491591822813806, −0.78128195290313643184655539802,
1.52508927705475606295631024807, 3.55625654872024008186287915727, 4.61348630462379721583122710740, 5.59436138106903164524974424256, 6.58389951402956851121229694009, 7.37331560848765146979901348062, 9.082792509294088292125919373568, 9.379737155928953315468836261591, 10.59414548102310923124725628797, 11.26045780397953008845513774433
