| L(s) = 1 | − 2.18·2-s − 2.69·3-s + 2.75·4-s + 0.125·5-s + 5.87·6-s − 3.80·7-s − 1.65·8-s + 4.24·9-s − 0.273·10-s − 6.52·11-s − 7.42·12-s − 4.53·13-s + 8.30·14-s − 0.337·15-s − 1.90·16-s + 6.34·17-s − 9.25·18-s − 19-s + 0.345·20-s + 10.2·21-s + 14.2·22-s + 0.121·23-s + 4.45·24-s − 4.98·25-s + 9.89·26-s − 3.34·27-s − 10.5·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 1.55·3-s + 1.37·4-s + 0.0560·5-s + 2.39·6-s − 1.43·7-s − 0.585·8-s + 1.41·9-s − 0.0864·10-s − 1.96·11-s − 2.14·12-s − 1.25·13-s + 2.22·14-s − 0.0870·15-s − 0.476·16-s + 1.53·17-s − 2.18·18-s − 0.229·19-s + 0.0772·20-s + 2.23·21-s + 3.03·22-s + 0.0253·23-s + 0.909·24-s − 0.996·25-s + 1.94·26-s − 0.644·27-s − 1.98·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 - 0.125T + 5T^{2} \) |
| 7 | \( 1 + 3.80T + 7T^{2} \) |
| 11 | \( 1 + 6.52T + 11T^{2} \) |
| 13 | \( 1 + 4.53T + 13T^{2} \) |
| 17 | \( 1 - 6.34T + 17T^{2} \) |
| 23 | \( 1 - 0.121T + 23T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 + 6.88T + 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 - 9.51T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 - 7.55T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 - 3.10T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 1.62T + 73T^{2} \) |
| 79 | \( 1 - 1.04T + 79T^{2} \) |
| 83 | \( 1 - 2.27T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.80579702270254138956553074651, −7.48929224126971133824609057386, −6.84449299090655305394629993493, −5.75579810156884718610027061002, −5.58164413579756778397426898662, −4.49954311109281366754552130004, −3.08522171803870673187945316342, −2.18274178870911612575087853191, −0.66331159608525547818317979136, 0,
0.66331159608525547818317979136, 2.18274178870911612575087853191, 3.08522171803870673187945316342, 4.49954311109281366754552130004, 5.58164413579756778397426898662, 5.75579810156884718610027061002, 6.84449299090655305394629993493, 7.48929224126971133824609057386, 7.80579702270254138956553074651
