| L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.104 + 0.994i)3-s + (0.309 − 0.951i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−2.78 + 0.592i)7-s + (−0.309 − 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.104 − 0.994i)10-s + (−2.70 − 3.00i)11-s + (0.913 + 0.406i)12-s + (−5.36 + 2.38i)13-s + (−1.90 + 2.11i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−5.07 + 5.63i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.0603 + 0.574i)3-s + (0.154 − 0.475i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (−1.05 + 0.224i)7-s + (−0.109 − 0.336i)8-s + (−0.326 − 0.0693i)9-s + (−0.0330 − 0.314i)10-s + (−0.815 − 0.905i)11-s + (0.263 + 0.117i)12-s + (−1.48 + 0.662i)13-s + (−0.509 + 0.566i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (−1.23 + 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00976064 + 0.0831201i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00976064 + 0.0831201i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-1.58 - 5.33i)T \) |
| good | 7 | \( 1 + (2.78 - 0.592i)T + (6.39 - 2.84i)T^{2} \) |
| 11 | \( 1 + (2.70 + 3.00i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (5.36 - 2.38i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (5.07 - 5.63i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.852 + 0.379i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (0.646 + 1.99i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-7.71 + 5.60i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (1.17 + 2.03i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.357 - 3.40i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (5.82 + 2.59i)T + (28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (-6.85 - 4.98i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (8.14 + 1.73i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.17 + 11.1i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 - 2.20T + 61T^{2} \) |
| 67 | \( 1 + (-3.87 + 6.70i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (14.1 + 3.00i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (3.02 + 3.36i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-0.739 + 0.820i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.689 - 6.55i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + (2.31 - 7.12i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.48 - 4.56i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.821430044526955415014131180163, −8.970216978796772637871442548212, −8.160971373674756902475465945969, −6.63361255693851849281180669560, −6.11493130561080044592451258468, −4.99709805765193607636687929242, −4.31740334097148050705638326137, −3.11888950837341902159144291043, −2.24261049425401868315376805542, −0.02890627873114934807648255740,
2.43092741868327566203372197811, 2.95766082046091327901450616321, 4.54018068238003678597281292707, 5.29943922728270095273565840806, 6.39774856482161757259992251527, 7.16730416876154004745830270214, 7.46710064707875587251563862217, 8.763785100525287037808212731731, 9.860175869902969631313865039167, 10.31288591891935013781487047796
