L(s) = 1 | + i·2-s + (−1.71 − 0.211i)3-s − 4-s + (0.584 − 1.01i)5-s + (0.211 − 1.71i)6-s − i·8-s + (2.91 + 0.725i)9-s + (1.01 + 0.584i)10-s + (−4.99 + 2.88i)11-s + (1.71 + 0.211i)12-s + (−0.571 + 0.329i)13-s + (−1.21 + 1.61i)15-s + 16-s + (2.83 − 4.91i)17-s + (−0.725 + 2.91i)18-s + (3.16 − 1.82i)19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.992 − 0.121i)3-s − 0.5·4-s + (0.261 − 0.453i)5-s + (0.0861 − 0.701i)6-s − 0.353i·8-s + (0.970 + 0.241i)9-s + (0.320 + 0.184i)10-s + (−1.50 + 0.869i)11-s + (0.496 + 0.0609i)12-s + (−0.158 + 0.0914i)13-s + (−0.314 + 0.417i)15-s + 0.250·16-s + (0.688 − 1.19i)17-s + (−0.171 + 0.686i)18-s + (0.726 − 0.419i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.782761 + 0.563395i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.782761 + 0.563395i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (1.71 + 0.211i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.584 + 1.01i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (4.99 - 2.88i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.571 - 0.329i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.83 + 4.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.16 + 1.82i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.503 + 0.290i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.53 - 3.77i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.17iT - 31T^{2} \) |
| 37 | \( 1 + (-5.29 - 9.17i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.64 - 4.57i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.37 + 4.12i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 + (4.14 + 2.39i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 1.02T + 59T^{2} \) |
| 61 | \( 1 - 11.7iT - 61T^{2} \) |
| 67 | \( 1 - 8.52T + 67T^{2} \) |
| 71 | \( 1 - 8.34iT - 71T^{2} \) |
| 73 | \( 1 + (0.899 + 0.519i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 1.26T + 79T^{2} \) |
| 83 | \( 1 + (6.21 - 10.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.83 + 13.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.5 - 7.26i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.05079182503123963876709467651, −9.701264127857342672433247027809, −8.476918684139690279891879292925, −7.42017478980370012983722946328, −7.04799864382237204767866329295, −5.81305622058764082768384080054, −5.02368027034372281120001989887, −4.70297795520873097637095860665, −2.82425755754221143942843006396, −1.02490009667103144809392885538,
0.68897127851867564487835918316, 2.34221156109408054384107132822, 3.48240924210925707291895725275, 4.62205071224589119809131316030, 5.70054883081947301641520447137, 6.09476179493271040627779100682, 7.54612240015156914105874778008, 8.213656012499755109840243266809, 9.545611582770298755526927338408, 10.30347815084963651208811390050