| L(s) = 1 | − 3·2-s − 5·4-s − 5-s + 14·7-s + 33·8-s + 3·10-s + 22·11-s − 25·13-s − 42·14-s − 21·16-s + 112·17-s − 71·19-s + 5·20-s − 66·22-s + 56·23-s − 211·25-s + 75·26-s − 70·28-s + 11·29-s − 310·31-s − 87·32-s − 336·34-s − 14·35-s − 65·37-s + 213·38-s − 33·40-s − 42·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 5/8·4-s − 0.0894·5-s + 0.755·7-s + 1.45·8-s + 0.0948·10-s + 0.603·11-s − 0.533·13-s − 0.801·14-s − 0.328·16-s + 1.59·17-s − 0.857·19-s + 0.0559·20-s − 0.639·22-s + 0.507·23-s − 1.68·25-s + 0.565·26-s − 0.472·28-s + 0.0704·29-s − 1.79·31-s − 0.480·32-s − 1.69·34-s − 0.0676·35-s − 0.288·37-s + 0.909·38-s − 0.130·40-s − 0.159·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_4$ | \( 1 + 3 T + 7 p T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 212 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 25 T + 3832 T^{2} + 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 112 T + 12894 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 71 T + 670 p T^{2} + 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 56 T + 24846 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 30852 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 p T + 80734 T^{2} + 10 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 65 T + 53704 T^{2} + 65 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 42 T + 106850 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 32 T + 98070 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 101 T + 97278 T^{2} + 101 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 166 T + 290346 T^{2} + 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 398850 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 436 T + 474286 T^{2} + 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 127 T + 567202 T^{2} + 127 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 936 T + 921518 T^{2} + 936 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 327 T + 377688 T^{2} + 327 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 228 T + 933726 T^{2} + 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 262 T + 1156910 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 44 T + 1212134 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2266 T + 3107658 T^{2} + 2266 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.579425032450930458840847296578, −9.531035240868883419639633291960, −8.968459757617524801808675741112, −8.685274798969743728111074218185, −8.053926276143849475536081811005, −7.920439516533049504713021588117, −7.38875864869095962304574483612, −7.09458888031229806519488101139, −6.22981283637926550830252299727, −5.82932443745915256131683548196, −5.15031104586740018230284077222, −4.97061842491356212840182660399, −4.03362215639515154882147189413, −4.03273550061096967927586777432, −3.21851181291526248165313877429, −2.35792862686512919898333997571, −1.38836932986094169123160992142, −1.37622371514551312910643013372, 0, 0,
1.37622371514551312910643013372, 1.38836932986094169123160992142, 2.35792862686512919898333997571, 3.21851181291526248165313877429, 4.03273550061096967927586777432, 4.03362215639515154882147189413, 4.97061842491356212840182660399, 5.15031104586740018230284077222, 5.82932443745915256131683548196, 6.22981283637926550830252299727, 7.09458888031229806519488101139, 7.38875864869095962304574483612, 7.920439516533049504713021588117, 8.053926276143849475536081811005, 8.685274798969743728111074218185, 8.968459757617524801808675741112, 9.531035240868883419639633291960, 9.579425032450930458840847296578
