L(s) = 1 | + 2.15·2-s + 2.63·4-s + 4.16·5-s − 7-s + 1.36·8-s + 8.96·10-s − 0.785·11-s + 13-s − 2.15·14-s − 2.32·16-s + 5.87·17-s − 8.59·19-s + 10.9·20-s − 1.69·22-s + 0.928·23-s + 12.3·25-s + 2.15·26-s − 2.63·28-s − 5.23·29-s − 5.32·31-s − 7.74·32-s + 12.6·34-s − 4.16·35-s − 1.27·37-s − 18.5·38-s + 5.69·40-s − 1.07·41-s + ⋯ |
L(s) = 1 | + 1.52·2-s + 1.31·4-s + 1.86·5-s − 0.377·7-s + 0.483·8-s + 2.83·10-s − 0.236·11-s + 0.277·13-s − 0.575·14-s − 0.581·16-s + 1.42·17-s − 1.97·19-s + 2.45·20-s − 0.360·22-s + 0.193·23-s + 2.46·25-s + 0.422·26-s − 0.497·28-s − 0.972·29-s − 0.956·31-s − 1.36·32-s + 2.16·34-s − 0.703·35-s − 0.208·37-s − 3.00·38-s + 0.899·40-s − 0.167·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.285179733\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.285179733\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 11 | \( 1 + 0.785T + 11T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 + 8.59T + 19T^{2} \) |
| 23 | \( 1 - 0.928T + 23T^{2} \) |
| 29 | \( 1 + 5.23T + 29T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 + 1.27T + 37T^{2} \) |
| 41 | \( 1 + 1.07T + 41T^{2} \) |
| 43 | \( 1 - 8.59T + 43T^{2} \) |
| 47 | \( 1 + 1.71T + 47T^{2} \) |
| 53 | \( 1 - 6.80T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 0.729T + 67T^{2} \) |
| 71 | \( 1 + 7.53T + 71T^{2} \) |
| 73 | \( 1 + 7.32T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−10.39391950162825509838884677361, −9.501756769675954805599238461766, −8.729556484779056728652069235537, −7.20845073705241882771261890257, −6.21411460785360391894893003926, −5.81388495874292836905526688715, −5.06885941335122762723816271960, −3.87700441564663722252474182920, −2.76846756556534370420167586187, −1.84619815599284622504893426267,
1.84619815599284622504893426267, 2.76846756556534370420167586187, 3.87700441564663722252474182920, 5.06885941335122762723816271960, 5.81388495874292836905526688715, 6.21411460785360391894893003926, 7.20845073705241882771261890257, 8.729556484779056728652069235537, 9.501756769675954805599238461766, 10.39391950162825509838884677361