| L(s) = 1 | + (82.4 − 142. i)3-s + (1.03e3 + 1.78e3i)5-s + (−6.31e3 + 658. i)7-s + (−3.75e3 − 6.50e3i)9-s + (1.95e4 − 3.39e4i)11-s − 1.43e4·13-s + 3.39e5·15-s + (−1.17e5 + 2.03e5i)17-s + (2.71e5 + 4.70e5i)19-s + (−4.26e5 + 9.56e5i)21-s + (−5.76e5 − 9.97e5i)23-s + (−1.14e6 + 1.98e6i)25-s + 2.00e6·27-s − 6.02e6·29-s + (−4.27e6 + 7.40e6i)31-s + ⋯ |
| L(s) = 1 | + (0.587 − 1.01i)3-s + (0.737 + 1.27i)5-s + (−0.994 + 0.103i)7-s + (−0.190 − 0.330i)9-s + (0.403 − 0.698i)11-s − 0.139·13-s + 1.73·15-s + (−0.341 + 0.591i)17-s + (0.478 + 0.829i)19-s + (−0.479 + 1.07i)21-s + (−0.429 − 0.743i)23-s + (−0.587 + 1.01i)25-s + 0.726·27-s − 1.58·29-s + (−0.831 + 1.44i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0404 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0404 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.707318407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707318407\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (6.31e3 - 658. i)T \) |
| good | 3 | \( 1 + (-82.4 + 142. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.03e3 - 1.78e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-1.95e4 + 3.39e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.43e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (1.17e5 - 2.03e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.71e5 - 4.70e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (5.76e5 + 9.97e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 6.02e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (4.27e6 - 7.40e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-2.47e6 - 4.28e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.48e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.80e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.38e7 - 4.12e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (4.94e7 - 8.56e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-2.07e7 + 3.59e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-1.71e7 - 2.96e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-7.66e7 + 1.32e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 1.59e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (2.03e8 - 3.52e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-4.27e7 - 7.41e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 5.27e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-2.67e8 - 4.62e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.26e9T + 7.60e17T^{2} \) |
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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
−12.40391282924384208158520727279, −10.94906342113585698453332781797, −10.02038108443399653804628660675, −8.870300077245126444012571021539, −7.55023194862818629525435489136, −6.61615924777429386657275217766, −5.92618675315577631945282400845, −3.51073248003331100923661022005, −2.60325130720949880223703549937, −1.50124852460389049820602399876,
0.37150723415457877223369647462, 1.99772537727437443533035094575, 3.54207076583581833798094716586, 4.59082149012047946116864987866, 5.68502686544385425293367636940, 7.21616901836900644382258526486, 8.911217934622053025142133986979, 9.471838977875973347374700266205, 9.925874010709423015848919389441, 11.60212465482272611120494249006
