| L(s) = 1 | + 0.879·2-s − 1.22·4-s − 3.87·5-s − 2.18·7-s − 2.83·8-s − 3.41·10-s − 0.162·11-s + 2.41·13-s − 1.92·14-s − 0.0418·16-s − 3·17-s + 3.59·19-s + 4.75·20-s − 0.142·22-s − 2.83·23-s + 10.0·25-s + 2.12·26-s + 2.68·28-s − 6.71·29-s − 5.18·31-s + 5.63·32-s − 2.63·34-s + 8.47·35-s − 6.63·37-s + 3.16·38-s + 11.0·40-s + 5.80·41-s + ⋯ |
| L(s) = 1 | + 0.621·2-s − 0.613·4-s − 1.73·5-s − 0.825·7-s − 1.00·8-s − 1.07·10-s − 0.0489·11-s + 0.668·13-s − 0.513·14-s − 0.0104·16-s − 0.727·17-s + 0.825·19-s + 1.06·20-s − 0.0304·22-s − 0.591·23-s + 2.00·25-s + 0.415·26-s + 0.506·28-s − 1.24·29-s − 0.931·31-s + 0.996·32-s − 0.452·34-s + 1.43·35-s − 1.09·37-s + 0.513·38-s + 1.74·40-s + 0.905·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| good | 2 | \( 1 - 0.879T + 2T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 + 0.162T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 + 2.83T + 23T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 + 6.63T + 37T^{2} \) |
| 41 | \( 1 - 5.80T + 41T^{2} \) |
| 43 | \( 1 + 6.22T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 + 3.78T + 61T^{2} \) |
| 67 | \( 1 + 5.86T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 8.68T + 73T^{2} \) |
| 79 | \( 1 + 1.26T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 + 7.72T + 89T^{2} \) |
| 97 | \( 1 + 3.90T + 97T^{2} \) |
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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
−11.79202620701691454481874163610, −10.99028265917821185663419975035, −9.563348264986510895218878526500, −8.651597628554224859418972476930, −7.68569108783688244384418038570, −6.51730807411975266096373473957, −5.15908875112276506092503140885, −3.91384805902053872584527872735, −3.38748189386890677398618911584, 0,
3.38748189386890677398618911584, 3.91384805902053872584527872735, 5.15908875112276506092503140885, 6.51730807411975266096373473957, 7.68569108783688244384418038570, 8.651597628554224859418972476930, 9.563348264986510895218878526500, 10.99028265917821185663419975035, 11.79202620701691454481874163610
