| L(s) = 1 | + 2·2-s − 4-s + 4·5-s − 8·8-s + 8·10-s + 4·11-s + 6·13-s − 7·16-s + 2·17-s − 4·19-s − 4·20-s + 8·22-s + 12·23-s + 11·25-s + 12·26-s − 8·31-s + 14·32-s + 4·34-s − 8·38-s − 32·40-s + 10·41-s − 8·43-s − 4·44-s + 24·46-s − 2·49-s + 22·50-s − 6·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 1.78·5-s − 2.82·8-s + 2.52·10-s + 1.20·11-s + 1.66·13-s − 7/4·16-s + 0.485·17-s − 0.917·19-s − 0.894·20-s + 1.70·22-s + 2.50·23-s + 11/5·25-s + 2.35·26-s − 1.43·31-s + 2.47·32-s + 0.685·34-s − 1.29·38-s − 5.05·40-s + 1.56·41-s − 1.21·43-s − 0.603·44-s + 3.53·46-s − 2/7·49-s + 3.11·50-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.199848638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.199848638\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.96173441549621109055244490487, −10.64157954044242576066951716427, −9.855024800371297737413440342235, −9.470124369939999959618787702666, −9.165252757339940547689775878091, −8.916867600522931754251102174121, −8.594865004276335029924990377878, −7.955795278386500156380358196704, −6.91665501908896555452861470523, −6.60632926791258043434031423381, −6.11493550004749757094352038986, −5.84578963802541326271123862655, −5.37486235754729167422464010863, −4.92579163984193927684428744318, −4.37935492157426664736521006447, −3.89937271997327476002133720248, −3.09361438745584696382400953568, −3.05973408863458077001242866713, −1.73607149997674596619277865121, −1.07254003322553323477080206566,
1.07254003322553323477080206566, 1.73607149997674596619277865121, 3.05973408863458077001242866713, 3.09361438745584696382400953568, 3.89937271997327476002133720248, 4.37935492157426664736521006447, 4.92579163984193927684428744318, 5.37486235754729167422464010863, 5.84578963802541326271123862655, 6.11493550004749757094352038986, 6.60632926791258043434031423381, 6.91665501908896555452861470523, 7.955795278386500156380358196704, 8.594865004276335029924990377878, 8.916867600522931754251102174121, 9.165252757339940547689775878091, 9.470124369939999959618787702666, 9.855024800371297737413440342235, 10.64157954044242576066951716427, 10.96173441549621109055244490487
