| L(s) = 1 | + (0.882 + 1.49i)3-s + (1.89 + 1.58i)5-s + (2.25 + 0.819i)7-s + (−1.44 + 2.63i)9-s + (−0.591 + 0.496i)11-s + (0.133 − 0.755i)13-s + (−0.696 + 4.22i)15-s + (3.04 − 5.27i)17-s + (2.64 + 4.58i)19-s + (0.766 + 4.07i)21-s + (−0.689 + 0.251i)23-s + (0.193 + 1.09i)25-s + (−5.19 + 0.175i)27-s + (−0.0924 − 0.524i)29-s + (−8.58 + 3.12i)31-s + ⋯ |
| L(s) = 1 | + (0.509 + 0.860i)3-s + (0.847 + 0.710i)5-s + (0.850 + 0.309i)7-s + (−0.480 + 0.877i)9-s + (−0.178 + 0.149i)11-s + (0.0369 − 0.209i)13-s + (−0.179 + 1.09i)15-s + (0.738 − 1.27i)17-s + (0.607 + 1.05i)19-s + (0.167 + 0.889i)21-s + (−0.143 + 0.0523i)23-s + (0.0387 + 0.219i)25-s + (−0.999 + 0.0337i)27-s + (−0.0171 − 0.0974i)29-s + (−1.54 + 0.560i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0825 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0825 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68430 + 1.55054i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68430 + 1.55054i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.882 - 1.49i)T \) |
| good | 5 | \( 1 + (-1.89 - 1.58i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.25 - 0.819i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (0.591 - 0.496i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.133 + 0.755i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.04 + 5.27i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.64 - 4.58i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.689 - 0.251i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.0924 + 0.524i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (8.58 - 3.12i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.587 + 1.01i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.88 + 10.6i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-5.29 + 4.44i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (4.78 + 1.74i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 9.59T + 53T^{2} \) |
| 59 | \( 1 + (-7.85 - 6.59i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-4.38 - 1.59i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.191 + 1.08i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (4.00 - 6.93i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.53 - 11.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.56 - 14.5i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.03 + 11.5i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (2.82 + 4.88i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.72 - 2.28i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.20510800939751105958647843785, −9.675373963726581416128104109701, −8.820331003647122019504401490442, −7.87786740636921458385555281724, −7.10961470603888390157303179945, −5.55350276099946866393040294904, −5.33344153673525461972348166812, −3.92707956784317352071719491191, −2.87831738359548264384062891206, −1.91550562258165883273620643189,
1.17937764038489336497371901857, 1.97825854974760242484844537866, 3.35962036248277441178083068551, 4.70323523799061938590663931180, 5.66008343638168549655609193979, 6.47883675641235367176660647855, 7.67562988348591603671353141241, 8.119164209236049865834981775674, 9.137558306645380192716949972297, 9.658471582893289787813275130536
