| L(s) = 1 | + (−29.1 + 50.5i)3-s + (2.15e3 − 1.24e3i)5-s + (2.21e3 − 5.95e3i)7-s + (8.13e3 + 1.40e4i)9-s + (7.19e4 + 4.15e4i)11-s + 8.97e4i·13-s + 1.45e5i·15-s + (−3.29e5 − 1.90e5i)17-s + (5.17e5 + 8.97e5i)19-s + (2.36e5 + 2.85e5i)21-s + (1.03e6 − 5.97e5i)23-s + (2.11e6 − 3.65e6i)25-s − 2.09e6·27-s − 3.28e6·29-s + (−2.15e5 + 3.73e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.208 + 0.360i)3-s + (1.54 − 0.889i)5-s + (0.348 − 0.937i)7-s + (0.413 + 0.716i)9-s + (1.48 + 0.855i)11-s + 0.871i·13-s + 0.739i·15-s + (−0.957 − 0.552i)17-s + (0.911 + 1.57i)19-s + (0.265 + 0.320i)21-s + (0.771 − 0.445i)23-s + (1.08 − 1.87i)25-s − 0.760·27-s − 0.861·29-s + (−0.0419 + 0.0726i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.272051535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.272051535\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.21e3 + 5.95e3i)T \) |
| good | 3 | \( 1 + (29.1 - 50.5i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-2.15e3 + 1.24e3i)T + (9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-7.19e4 - 4.15e4i)T + (1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 - 8.97e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (3.29e5 + 1.90e5i)T + (5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-5.17e5 - 8.97e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.03e6 + 5.97e5i)T + (9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + 3.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (2.15e5 - 3.73e5i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.55e6 - 2.70e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 6.97e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 1.88e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.26e7 - 3.91e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-9.74e5 + 1.68e6i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (6.20e7 - 1.07e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-2.30e7 + 1.33e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (7.12e7 + 4.11e7i)T + (1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 7.73e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (1.35e8 + 7.79e7i)T + (2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-4.27e8 + 2.46e8i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.68e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-2.58e8 + 1.49e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.65e9iT - 7.60e17T^{2} \) |
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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
−11.90451638005013205598601280520, −10.64552432905698180217455758479, −9.652147050010069124949214983765, −9.114351719440055131221569080894, −7.40621358538738486206546233529, −6.25117541080210053685217337958, −4.90286201183896084738473821369, −4.21144690782478476204590343181, −1.85246384967513785724382531746, −1.27270829643919295164726232523,
0.980486638799850890511241013490, 2.13233881450518511438491731007, 3.34859708197826392050081197421, 5.40227856249824642970299686529, 6.25007484238052812744219077153, 7.00378972336589688241567566501, 8.962786265544541200364279639344, 9.446907522143293787891346494545, 10.88652112411656873659284039886, 11.65758440886438467444322905529
