| L(s) = 1 | − 2·2-s − 2·3-s − 10·4-s + 2·5-s + 4·6-s − 20·7-s + 32·8-s − 3·9-s − 4·10-s + 20·12-s − 80·13-s + 40·14-s − 4·15-s + 44·16-s + 124·17-s + 6·18-s − 72·19-s − 20·20-s + 40·21-s − 98·23-s − 64·24-s − 55·25-s + 160·26-s − 34·27-s + 200·28-s − 144·29-s + 8·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.384·3-s − 5/4·4-s + 0.178·5-s + 0.272·6-s − 1.07·7-s + 1.41·8-s − 1/9·9-s − 0.126·10-s + 0.481·12-s − 1.70·13-s + 0.763·14-s − 0.0688·15-s + 0.687·16-s + 1.76·17-s + 0.0785·18-s − 0.869·19-s − 0.223·20-s + 0.415·21-s − 0.888·23-s − 0.544·24-s − 0.439·25-s + 1.20·26-s − 0.242·27-s + 1.34·28-s − 0.922·29-s + 0.0486·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 7 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 59 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 738 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 4794 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 124 T + 13238 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 72 T + 4214 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 98 T + 22847 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 144 T + 44554 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 34 T + 57519 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 54 T + 101843 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 536 T + 209618 T^{2} + 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 60 T + 159146 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 272 T + 182942 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 492 T + 348862 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 634 T + 458975 T^{2} - 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 840 T + 528794 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 754 T + 742455 T^{2} - 754 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 678 T + 813415 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 400 T + 160962 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 p T - 279966 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 468 T + 1155130 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1842 T + 1935427 T^{2} + 1842 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2194 T + 2966547 T^{2} - 2194 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.60049138921098189658558253409, −12.39583005398207801588432958358, −11.80894701207969181601106078217, −11.06214396111645016321095234316, −10.22117977230772887434677405655, −9.901678808544750095756864408089, −9.661350687673465733032615363812, −9.305897447054092520175674310873, −8.295524804042286583231505106990, −8.095185549472386784858743798541, −7.35543080829368817035264404875, −6.68386104755497933577516822007, −5.82553932576668162270287513298, −5.32280648645674888756322103045, −4.65150366772600906780035440712, −3.83879729854836863296002502972, −3.05088622613104236600227662442, −1.66692501185640243428159567255, 0, 0,
1.66692501185640243428159567255, 3.05088622613104236600227662442, 3.83879729854836863296002502972, 4.65150366772600906780035440712, 5.32280648645674888756322103045, 5.82553932576668162270287513298, 6.68386104755497933577516822007, 7.35543080829368817035264404875, 8.095185549472386784858743798541, 8.295524804042286583231505106990, 9.305897447054092520175674310873, 9.661350687673465733032615363812, 9.901678808544750095756864408089, 10.22117977230772887434677405655, 11.06214396111645016321095234316, 11.80894701207969181601106078217, 12.39583005398207801588432958358, 12.60049138921098189658558253409
