| L(s) = 1 | + (14.9 − 5.69i)2-s + (−106. − 34.4i)3-s + (191. − 170. i)4-s + (−548. − 299. i)5-s + (−1.78e3 + 88.4i)6-s − 2.48e3i·7-s + (1.89e3 − 3.63e3i)8-s + (4.77e3 + 3.46e3i)9-s + (−9.90e3 − 1.36e3i)10-s + (−2.57e3 − 3.53e3i)11-s + (−2.61e4 + 1.14e4i)12-s + (−3.13e4 − 2.27e4i)13-s + (−1.41e4 − 3.72e4i)14-s + (4.78e4 + 5.07e4i)15-s + (7.58e3 − 6.50e4i)16-s + (−4.52e4 − 1.39e5i)17-s + ⋯ |
| L(s) = 1 | + (0.934 − 0.355i)2-s + (−1.31 − 0.425i)3-s + (0.746 − 0.664i)4-s + (−0.877 − 0.479i)5-s + (−1.37 + 0.0682i)6-s − 1.03i·7-s + (0.461 − 0.887i)8-s + (0.727 + 0.528i)9-s + (−0.990 − 0.136i)10-s + (−0.175 − 0.241i)11-s + (−1.26 + 0.553i)12-s + (−1.09 − 0.797i)13-s + (−0.368 − 0.968i)14-s + (0.945 + 1.00i)15-s + (0.115 − 0.993i)16-s + (−0.541 − 1.66i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.593173 + 0.701355i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.593173 + 0.701355i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-14.9 + 5.69i)T \) |
| 5 | \( 1 + (548. + 299. i)T \) |
| good | 3 | \( 1 + (106. + 34.4i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 + 2.48e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (2.57e3 + 3.53e3i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (3.13e4 + 2.27e4i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (4.52e4 + 1.39e5i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-8.88e4 + 2.88e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (-1.08e5 - 1.49e5i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (2.24e5 - 6.89e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-6.89e5 + 2.24e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (1.74e6 + 1.27e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-2.79e6 - 2.03e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 5.12e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (6.01e6 + 1.95e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (-5.98e5 + 1.84e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-9.06e6 + 1.24e7i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (5.55e6 - 4.03e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-1.70e7 + 5.54e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-3.15e7 - 1.02e7i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-2.15e7 + 1.56e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (6.95e7 + 2.26e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (5.05e7 - 1.64e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (6.82e6 - 4.95e6i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (-3.52e7 + 1.08e8i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−11.50017588451729237479114325300, −11.08216873651541441648934840029, −9.768178990192207571593335722924, −7.51547399220374782366707680038, −6.90559195996212287182075383333, −5.28772297848039620305399559188, −4.71356596296838085911198744608, −3.14013623240132555439801815501, −0.970921947804729547303067365296, −0.27481357264462475766195993798,
2.37803646911140275533911926022, 4.03316308535931877887683847093, 5.00704542271781526841239028911, 6.07184010989398664896340946276, 7.03078588707039961440269656249, 8.413697063910736390455550667257, 10.27872126277298518336636705834, 11.32073607321208826474015842646, 12.00537876064779765234199667998, 12.58272810590051392909253449944
