| L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−1.94 + 3.37i)5-s + 7.99·8-s + (−3.89 − 6.75i)10-s + (−30.6 − 53.1i)11-s + 53.6·13-s + (−8 + 13.8i)16-s + (16.0 + 27.7i)17-s + (27.8 − 48.3i)19-s + 15.5·20-s + 122.·22-s + (−47.3 + 81.9i)23-s + (54.8 + 95.0i)25-s + (−53.6 + 93.0i)26-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.174 + 0.302i)5-s + 0.353·8-s + (−0.123 − 0.213i)10-s + (−0.841 − 1.45i)11-s + 1.14·13-s + (−0.125 + 0.216i)16-s + (0.228 + 0.396i)17-s + (0.336 − 0.583i)19-s + 0.174·20-s + 1.18·22-s + (−0.428 + 0.742i)23-s + (0.439 + 0.760i)25-s + (−0.405 + 0.701i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1185601078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1185601078\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (1.94 - 3.37i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (30.6 + 53.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 53.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-16.0 - 27.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-27.8 + 48.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (47.3 - 81.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (66.3 + 114. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (74.6 - 129. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 427.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 437.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-28.5 + 49.3i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (131. + 228. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-225. - 391. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (289. - 501. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (154. + 268. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (596. + 1.03e3i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (659. - 1.14e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (116. - 201. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.60e3T + 9.12e5T^{2} \) |
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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
−9.213943150564752832197957797161, −8.493704418026187033364146519705, −7.80716761397182930588666868817, −6.92499537899550202881687012256, −5.85557896214723978667630858066, −5.41876922584371185161142576171, −3.89742578170620077414058301801, −3.02345394601714706089783651225, −1.34940915246890359347865127888, −0.03686373084440719033727888176,
1.38416535015400387237811320983, 2.44628586350028678521662904842, 3.67868424635540567684310758534, 4.60344459027258353948783834151, 5.56067695812862424212120125458, 6.84044259713309432747070413720, 7.72356872545223448914462661302, 8.454361363516579484787587962573, 9.320377571425699083178068445611, 10.16863479477139264048412644698
