| L(s) = 1 | + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + (2.29 + 3.97i)5-s − 7.99·8-s + (−4.58 + 7.94i)10-s + (−3.24 + 5.61i)11-s + 45.2·13-s + (−8 − 13.8i)16-s + (40.7 − 70.6i)17-s + (2.52 + 4.37i)19-s − 18.3·20-s − 12.9·22-s + (53.1 + 92.0i)23-s + (51.9 − 90.0i)25-s + (45.2 + 78.3i)26-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.205 + 0.355i)5-s − 0.353·8-s + (−0.145 + 0.251i)10-s + (−0.0888 + 0.153i)11-s + 0.964·13-s + (−0.125 − 0.216i)16-s + (0.581 − 1.00i)17-s + (0.0305 + 0.0528i)19-s − 0.205·20-s − 0.125·22-s + (0.481 + 0.834i)23-s + (0.415 − 0.720i)25-s + (0.341 + 0.590i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.636524712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.636524712\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-2.29 - 3.97i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (3.24 - 5.61i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 45.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-40.7 + 70.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-2.52 - 4.37i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-53.1 - 92.0i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (146. - 253. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (57.2 + 99.2i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 161.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 471.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-173. - 299. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-202. + 351. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (126. - 219. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-375. - 650. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (5.82 - 10.0i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (342. - 593. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (0.132 + 0.228i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 437.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (29.2 + 50.6i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.28e3T + 9.12e5T^{2} \) |
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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
−9.954428232264865103131789188769, −8.958444384938561733665832509437, −8.253377055709105847747109135807, −7.19763933558464339906047412903, −6.61892309873185445637982775011, −5.60154878452368981526220408082, −4.83390883946067356174364271003, −3.62174416181199247041007790572, −2.73138084125409056618180442270, −1.08230766597919244876640263595,
0.73444805336009011881767031159, 1.78800820376696563423529087367, 3.09041863033130340418872448228, 4.00214136728707687043970936329, 5.03295910020243852627312130830, 5.90668238405738728386918513130, 6.73961396580625958963195872834, 8.116705033778733180675554735628, 8.716210526599749541907754019943, 9.651937301472687439371853484095
