| L(s) = 1 | + 0.347·2-s + 0.879·3-s − 1.87·4-s − 2.34·5-s + 0.305·6-s − 1.87·7-s − 1.34·8-s − 2.22·9-s − 0.815·10-s − 5.06·11-s − 1.65·12-s + 4.71·13-s − 0.652·14-s − 2.06·15-s + 3.29·16-s − 0.773·18-s + 0.347·19-s + 4.41·20-s − 1.65·21-s − 1.75·22-s − 1.77·23-s − 1.18·24-s + 0.509·25-s + 1.63·26-s − 4.59·27-s + 3.53·28-s + 2.22·29-s + ⋯ |
| L(s) = 1 | + 0.245·2-s + 0.507·3-s − 0.939·4-s − 1.04·5-s + 0.124·6-s − 0.710·7-s − 0.476·8-s − 0.742·9-s − 0.257·10-s − 1.52·11-s − 0.477·12-s + 1.30·13-s − 0.174·14-s − 0.532·15-s + 0.822·16-s − 0.182·18-s + 0.0796·19-s + 0.986·20-s − 0.360·21-s − 0.374·22-s − 0.369·23-s − 0.241·24-s + 0.101·25-s + 0.321·26-s − 0.884·27-s + 0.667·28-s + 0.413·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 - 0.347T + 2T^{2} \) |
| 3 | \( 1 - 0.879T + 3T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 + 5.06T + 11T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 19 | \( 1 - 0.347T + 19T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 + 1.47T + 43T^{2} \) |
| 47 | \( 1 + 8.53T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 5.00T + 59T^{2} \) |
| 61 | \( 1 + 0.184T + 61T^{2} \) |
| 67 | \( 1 + 2.44T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 4.43T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 6.32T + 89T^{2} \) |
| 97 | \( 1 - 9.27T + 97T^{2} \) |
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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
−11.39539038219369690357962180383, −10.37931891707767172642975436250, −9.276917454792210431754159329685, −8.286701486524083253882570855504, −7.911005858675175450149494303552, −6.22352672939552447266743952757, −5.10039387682638716031480848279, −3.78447695175709354390604998397, −3.03900678633991776810654219890, 0,
3.03900678633991776810654219890, 3.78447695175709354390604998397, 5.10039387682638716031480848279, 6.22352672939552447266743952757, 7.911005858675175450149494303552, 8.286701486524083253882570855504, 9.276917454792210431754159329685, 10.37931891707767172642975436250, 11.39539038219369690357962180383
