| L(s) = 1 | − 37.4·2-s + 70.6·3-s + 890.·4-s + 613.·5-s − 2.64e3·6-s − 6.61e3·7-s − 1.41e4·8-s − 1.46e4·9-s − 2.29e4·10-s − 1.46e4·11-s + 6.29e4·12-s − 2.33e4·13-s + 2.47e5·14-s + 4.33e4·15-s + 7.50e4·16-s − 3.73e5·17-s + 5.50e5·18-s − 9.74e5·19-s + 5.46e5·20-s − 4.67e5·21-s + 5.48e5·22-s + 2.43e6·23-s − 1.00e6·24-s − 1.57e6·25-s + 8.75e5·26-s − 2.42e6·27-s − 5.88e6·28-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 0.503·3-s + 1.73·4-s + 0.439·5-s − 0.833·6-s − 1.04·7-s − 1.22·8-s − 0.746·9-s − 0.727·10-s − 0.301·11-s + 0.876·12-s − 0.227·13-s + 1.72·14-s + 0.221·15-s + 0.286·16-s − 1.08·17-s + 1.23·18-s − 1.71·19-s + 0.764·20-s − 0.524·21-s + 0.499·22-s + 1.81·23-s − 0.616·24-s − 0.807·25-s + 0.375·26-s − 0.879·27-s − 1.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 1.46e4T \) |
| good | 2 | \( 1 + 37.4T + 512T^{2} \) |
| 3 | \( 1 - 70.6T + 1.96e4T^{2} \) |
| 5 | \( 1 - 613.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.61e3T + 4.03e7T^{2} \) |
| 13 | \( 1 + 2.33e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.73e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.74e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.43e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.48e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.96e4T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.58e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.05e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.09e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.28e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.20e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.35e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.18e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.96e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.49e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.06e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.34e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.79e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.97e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.48e8T + 7.60e17T^{2} \) |
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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
−17.64466417770849373887276861200, −16.69422116113071368348011777849, −15.21054583605370965022929667075, −13.19022460811479344024129075614, −10.94995082471520698560852450286, −9.534650477708098070183386593283, −8.496898759074231993476957986874, −6.62401319865353759435682800448, −2.45730205558439381856307838203, 0,
2.45730205558439381856307838203, 6.62401319865353759435682800448, 8.496898759074231993476957986874, 9.534650477708098070183386593283, 10.94995082471520698560852450286, 13.19022460811479344024129075614, 15.21054583605370965022929667075, 16.69422116113071368348011777849, 17.64466417770849373887276861200
