| 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
−20.29031895492168728176722207886, −17.95941704426974621418917302378, −17.11713670221573348764625582796, −15.97237557396940374630601714750, −14.94871558638329035373088018842, −12.39372023266760880336534526809, −10.50904406057471169578365888840, −8.690535790972845654550945726360, −6.83901750814888482421220081455, −4.66616977776350690224064257483,
0.22301760536578800875679956916, 2.34636611506032936189145138441, 6.04610767527301754161709023430, 8.407181040429168640871681941960, 10.75827984950997250322471830419, 11.53041547659758222060810080481, 13.12791857506404652732046385606, 15.31861336998794459132681244094, 17.34254026432992826975471527857, 18.76026756234719257497234031171
