| L(s) = 1 | + 4·4-s + 13·5-s − 44·7-s + 44·13-s − 48·16-s + 99·17-s − 121·19-s + 52·20-s + 74·23-s + 28·25-s − 176·28-s − 132·29-s + 11·31-s − 572·35-s − 512·37-s − 88·41-s − 66·43-s − 345·47-s + 811·49-s + 176·52-s − 339·53-s − 385·59-s − 1.15e3·61-s − 448·64-s + 572·65-s + 75·67-s + 396·68-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.16·5-s − 2.37·7-s + 0.938·13-s − 3/4·16-s + 1.41·17-s − 1.46·19-s + 0.581·20-s + 0.670·23-s + 0.223·25-s − 1.18·28-s − 0.845·29-s + 0.0637·31-s − 2.76·35-s − 2.27·37-s − 0.335·41-s − 0.234·43-s − 1.07·47-s + 2.36·49-s + 0.469·52-s − 0.878·53-s − 0.849·59-s − 2.42·61-s − 7/8·64-s + 1.09·65-s + 0.136·67-s + 0.706·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 13 T + 141 T^{2} - 13 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 44 T + 1125 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 44 T + 4273 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 99 T + 8215 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 121 T + 14847 T^{2} + 121 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 74 T + 23283 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 132 T + 7054 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 11 T + 52201 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 512 T + 166237 T^{2} + 512 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 88 T + 138333 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 66 T + 151283 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 345 T + 233621 T^{2} + 345 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 339 T + 300923 T^{2} + 339 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 385 T + 193563 T^{2} + 385 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1155 T + 690167 T^{2} + 1155 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 75 T + 209531 T^{2} - 75 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1141 T + 946761 T^{2} + 1141 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 374 T + 686598 T^{2} - 374 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 572 T + 451869 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 396 T + 911333 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2712 T + 3219029 T^{2} - 2712 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 327 T + 859727 T^{2} + 327 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.185928386965953002688075878487, −9.067993520320030303006798169238, −8.718652267383389005254616496504, −8.056550625371943559287554835510, −7.42167221159423448859561894508, −7.11960457751909404597144542247, −6.54460847070156372085434030993, −6.20154279463819058172236592291, −6.17126176133304882598860117261, −5.71875541078202576131325168621, −4.97800079569468287258270270283, −4.57785114936185897649872144074, −3.60713431472788792342310317264, −3.29482976453783272211262926206, −3.18690677758877936975546169677, −2.28704326911949618974838210751, −1.82027018288169185474146520563, −1.25550013630007108647111073837, 0, 0,
1.25550013630007108647111073837, 1.82027018288169185474146520563, 2.28704326911949618974838210751, 3.18690677758877936975546169677, 3.29482976453783272211262926206, 3.60713431472788792342310317264, 4.57785114936185897649872144074, 4.97800079569468287258270270283, 5.71875541078202576131325168621, 6.17126176133304882598860117261, 6.20154279463819058172236592291, 6.54460847070156372085434030993, 7.11960457751909404597144542247, 7.42167221159423448859561894508, 8.056550625371943559287554835510, 8.718652267383389005254616496504, 9.067993520320030303006798169238, 9.185928386965953002688075878487
