| L(s) = 1 | + (−12.8 − 22.2i)2-s + (−90.7 − 106. i)3-s + (−73.0 + 126. i)4-s + (−353. + 611. i)5-s + (−1.21e3 + 3.38e3i)6-s + (2.28e3 + 3.95e3i)7-s − 9.38e3·8-s + (−3.19e3 + 1.94e4i)9-s + 1.81e4·10-s + (−3.57e4 − 6.18e4i)11-s + (2.01e4 − 3.67e3i)12-s + (1.32e4 − 2.28e4i)13-s + (5.85e4 − 1.01e5i)14-s + (9.75e4 − 1.77e4i)15-s + (1.57e5 + 2.73e5i)16-s − 3.14e5·17-s + ⋯ |
| L(s) = 1 | + (−0.566 − 0.981i)2-s + (−0.647 − 0.762i)3-s + (−0.142 + 0.247i)4-s + (−0.252 + 0.437i)5-s + (−0.381 + 1.06i)6-s + (0.359 + 0.622i)7-s − 0.810·8-s + (−0.162 + 0.986i)9-s + 0.573·10-s + (−0.735 − 1.27i)11-s + (0.280 − 0.0511i)12-s + (0.128 − 0.222i)13-s + (0.407 − 0.706i)14-s + (0.497 − 0.0906i)15-s + (0.601 + 1.04i)16-s − 0.914·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.139877 + 0.245538i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.139877 + 0.245538i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (90.7 + 106. i)T \) |
| good | 2 | \( 1 + (12.8 + 22.2i)T + (-256 + 443. i)T^{2} \) |
| 5 | \( 1 + (353. - 611. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (-2.28e3 - 3.95e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (3.57e4 + 6.18e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-1.32e4 + 2.28e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + 3.14e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + (-5.59e4 + 9.68e4i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (2.19e6 + 3.80e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-4.94e6 + 8.56e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 6.32e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (2.68e6 - 4.64e6i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.12e7 + 1.94e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-2.23e7 - 3.87e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + 4.00e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + (-4.79e7 + 8.29e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.01e7 + 1.75e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (9.12e7 - 1.58e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 1.38e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (1.97e8 + 3.41e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-2.36e8 - 4.09e8i)T + (-9.34e16 + 1.61e17i)T^{2} \) |
| 89 | \( 1 - 2.60e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (2.87e8 + 4.97e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−18.76026756234719257497234031171, −17.34254026432992826975471527857, −15.31861336998794459132681244094, −13.12791857506404652732046385606, −11.53041547659758222060810080481, −10.75827984950997250322471830419, −8.407181040429168640871681941960, −6.04610767527301754161709023430, −2.34636611506032936189145138441, −0.22301760536578800875679956916,
4.66616977776350690224064257483, 6.83901750814888482421220081455, 8.690535790972845654550945726360, 10.50904406057471169578365888840, 12.39372023266760880336534526809, 14.94871558638329035373088018842, 15.97237557396940374630601714750, 17.11713670221573348764625582796, 17.95941704426974621418917302378, 20.29031895492168728176722207886
