| L(s) = 1 | − 2.64·2-s − 0.999·4-s + 23.8·8-s − 26.4·11-s − 55.0·16-s + 70·22-s + 216.·23-s − 125·25-s + 264.·29-s − 44.9·32-s − 450·37-s + 180·43-s + 26.4·44-s − 574·46-s + 330.·50-s − 497.·53-s − 700.·58-s + 559·64-s − 740·67-s − 978.·71-s + 1.19e3·74-s − 1.38e3·79-s − 476.·86-s − 630·88-s − 216.·92-s + 124.·100-s + 1.31e3·106-s + ⋯ |
| L(s) = 1 | − 0.935·2-s − 0.124·4-s + 1.05·8-s − 0.725·11-s − 0.859·16-s + 0.678·22-s + 1.96·23-s − 25-s + 1.69·29-s − 0.248·32-s − 1.99·37-s + 0.638·43-s + 0.0906·44-s − 1.83·46-s + 0.935·50-s − 1.28·53-s − 1.58·58-s + 1.09·64-s − 1.34·67-s − 1.63·71-s + 1.87·74-s − 1.97·79-s − 0.597·86-s − 0.763·88-s − 0.245·92-s + 0.124·100-s + 1.20·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.64T + 8T^{2} \) |
| 5 | \( 1 + 125T^{2} \) |
| 11 | \( 1 + 26.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 - 216.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 264.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 450T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 180T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 497.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 2.26e5T^{2} \) |
| 67 | \( 1 + 740T + 3.00e5T^{2} \) |
| 71 | \( 1 + 978.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.38e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−10.26837571227162746484887336399, −9.246863196431183422737521567858, −8.560411296141730710991418894925, −7.68185555225961218857103254846, −6.81639193963138171549106642308, −5.37765930889433709071841057276, −4.46406743721194004553333993575, −2.94871509377801391643707825415, −1.38095841218195917023758122098, 0,
1.38095841218195917023758122098, 2.94871509377801391643707825415, 4.46406743721194004553333993575, 5.37765930889433709071841057276, 6.81639193963138171549106642308, 7.68185555225961218857103254846, 8.560411296141730710991418894925, 9.246863196431183422737521567858, 10.26837571227162746484887336399
