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
−11.26045780397953008845513774433, −10.59414548102310923124725628797, −9.379737155928953315468836261591, −9.082792509294088292125919373568, −7.37331560848765146979901348062, −6.58389951402956851121229694009, −5.59436138106903164524974424256, −4.61348630462379721583122710740, −3.55625654872024008186287915727, −1.52508927705475606295631024807,
0.78128195290313643184655539802, 2.60768499841721491591822813806, 4.19072717321892399566436110607, 5.42895126463658669857494231204, 6.21383006832052271347838414068, 7.10103287926095881016010827160, 8.097056802047865341061097350301, 9.227886316023637314521202799199, 10.36702283001576579059589793315, 10.94462797363261135967894380498