L(s) = 1 | − 2.61·2-s + 4.81·4-s + 3.81·5-s + 7-s − 7.34·8-s − 9.95·10-s + 4.73·11-s + 13-s − 2.61·14-s + 9.55·16-s + 5.22·17-s + 2.92·19-s + 18.3·20-s − 12.3·22-s − 3.33·23-s + 9.55·25-s − 2.61·26-s + 4.81·28-s + 0.922·29-s − 7.51·31-s − 10.2·32-s − 13.6·34-s + 3.81·35-s + 0.154·37-s − 7.62·38-s − 28.0·40-s − 6.36·41-s + ⋯ |
L(s) = 1 | − 1.84·2-s + 2.40·4-s + 1.70·5-s + 0.377·7-s − 2.59·8-s − 3.14·10-s + 1.42·11-s + 0.277·13-s − 0.697·14-s + 2.38·16-s + 1.26·17-s + 0.670·19-s + 4.10·20-s − 2.63·22-s − 0.694·23-s + 1.91·25-s − 0.511·26-s + 0.909·28-s + 0.171·29-s − 1.35·31-s − 1.81·32-s − 2.33·34-s + 0.644·35-s + 0.0254·37-s − 1.23·38-s − 4.43·40-s − 0.994·41-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
\(L(1)\) |
\(\approx\) |
\(1.092076102\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.092076102\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 + 3.33T + 23T^{2} \) |
| 29 | \( 1 - 0.922T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 - 0.154T + 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 + 9.03T + 47T^{2} \) |
| 53 | \( 1 + 8.55T + 53T^{2} \) |
| 59 | \( 1 + 3.95T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 + 7.73T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 1.40T + 83T^{2} \) |
| 89 | \( 1 - 1.96T + 89T^{2} \) |
| 97 | \( 1 + 2.11T + 97T^{2} \) |
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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
−9.832021551208454203385093285091, −9.560840795973092206260117704286, −8.781892567660636655992063339716, −7.926776299116200833459739847425, −6.86283475840592156110528526496, −6.22244172744608057430674102766, −5.33751356976887636667312193259, −3.30452041657188043684688722046, −1.84670722984431796738065311147, −1.32174650167891728778885897141,
1.32174650167891728778885897141, 1.84670722984431796738065311147, 3.30452041657188043684688722046, 5.33751356976887636667312193259, 6.22244172744608057430674102766, 6.86283475840592156110528526496, 7.926776299116200833459739847425, 8.781892567660636655992063339716, 9.560840795973092206260117704286, 9.832021551208454203385093285091