| L(s) = 1 | + (0.00266 − 0.00461i)2-s + (−12.1 − 21.0i)3-s + (63.9 + 110. i)4-s + (−249. + 432. i)5-s − 0.129·6-s + 1.36·8-s + (797. − 1.38e3i)9-s + (1.33 + 2.30i)10-s + (−2.59e3 − 4.49e3i)11-s + (1.55e3 − 2.69e3i)12-s − 3.04e3·13-s + 1.21e4·15-s + (−8.19e3 + 1.41e4i)16-s + (−1.02e4 − 1.77e4i)17-s + (−4.25 − 7.36i)18-s + (−1.77e4 + 3.06e4i)19-s + ⋯ |
| L(s) = 1 | + (0.000235 − 0.000408i)2-s + (−0.260 − 0.450i)3-s + (0.499 + 0.866i)4-s + (−0.892 + 1.54i)5-s − 0.000245·6-s + 0.000942·8-s + (0.364 − 0.631i)9-s + (0.000420 + 0.000728i)10-s + (−0.587 − 1.01i)11-s + (0.260 − 0.450i)12-s − 0.384·13-s + 0.928·15-s + (−0.499 + 0.866i)16-s + (−0.507 − 0.878i)17-s + (−0.000171 − 0.000297i)18-s + (−0.592 + 1.02i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0272744 - 0.118901i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0272744 - 0.118901i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (-0.00266 + 0.00461i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (12.1 + 21.0i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (249. - 432. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (2.59e3 + 4.49e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 3.04e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (1.02e4 + 1.77e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.77e4 - 3.06e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-9.26e3 + 1.60e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 1.43e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (9.77e4 + 1.69e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (1.36e5 - 2.36e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 1.81e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.96e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.18e5 + 5.52e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-7.15e5 - 1.23e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-3.19e5 - 5.53e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.05e5 - 1.83e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.15e6 + 1.99e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 7.06e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.93e6 - 3.34e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.89e6 - 3.29e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 1.41e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (1.01e6 - 1.75e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 9.70e6T + 8.07e13T^{2} \) |
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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
−14.96111445071373785364492364555, −13.53816686335362088341335733060, −12.14563875282485845172858928383, −11.46545170529740326980636654419, −10.41039046631091388759548078108, −8.254653744341109941691941293742, −7.20963013709298940843777918688, −6.41343767752637376162432785075, −3.75960502348983420627292038741, −2.66649976016705919642415887774,
0.04736434534445508186322084455, 1.75727163966473034838678134834, 4.53443961569247802341944057262, 5.15880066946376109026300250460, 7.16530802486492645740471226474, 8.596323087252277888115156604590, 9.959686052911305128887437551204, 11.01404644351378304903666255067, 12.26983433169768318632073185335, 13.23144142292925387812107742204
