| L(s) = 1 | + 2·2-s + 2·4-s + 4·8-s + 8·16-s − 8·31-s + 8·32-s + 8·37-s + 8·41-s − 28·43-s + 20·49-s − 32·53-s − 16·62-s + 8·64-s − 36·67-s − 8·71-s + 16·74-s + 32·79-s + 16·82-s − 12·83-s − 56·86-s + 8·89-s + 40·98-s − 64·106-s + 4·107-s + 36·121-s − 16·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.41·8-s + 2·16-s − 1.43·31-s + 1.41·32-s + 1.31·37-s + 1.24·41-s − 4.26·43-s + 20/7·49-s − 4.39·53-s − 2.03·62-s + 64-s − 4.39·67-s − 0.949·71-s + 1.85·74-s + 3.60·79-s + 1.76·82-s − 1.31·83-s − 6.03·86-s + 0.847·89-s + 4.04·98-s − 6.21·106-s + 0.386·107-s + 3.27·121-s − 1.43·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.167048193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.167048193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1274 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 132 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 23814 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 172 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 16806 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.65058792096698055138610119179, −6.10198838779237286729808582675, −6.04001871966838415054976521030, −5.99714360581013475599354408284, −5.87446975762017910099718519072, −5.52333952273113681477491645753, −5.22963594832064517190308341869, −4.98132991282355606147251205535, −4.80934586604322857548505472528, −4.61159888714951671126957877199, −4.57003982773389619526367057303, −4.22611767938686623024153711150, −4.13382248707714998064994717799, −3.51061855089185071054114982131, −3.47213425367116401917644493260, −3.37619777175556280002289084970, −3.20731227091080207935238350638, −2.79837211638562991581086376321, −2.44121191902638750255597298629, −2.16408368685257094003247646809, −1.88145470132716592893854267579, −1.57985149051506913254164038015, −1.32545156543905074664538973758, −0.983054897969617467695536149421, −0.18164856591516524772330564078,
0.18164856591516524772330564078, 0.983054897969617467695536149421, 1.32545156543905074664538973758, 1.57985149051506913254164038015, 1.88145470132716592893854267579, 2.16408368685257094003247646809, 2.44121191902638750255597298629, 2.79837211638562991581086376321, 3.20731227091080207935238350638, 3.37619777175556280002289084970, 3.47213425367116401917644493260, 3.51061855089185071054114982131, 4.13382248707714998064994717799, 4.22611767938686623024153711150, 4.57003982773389619526367057303, 4.61159888714951671126957877199, 4.80934586604322857548505472528, 4.98132991282355606147251205535, 5.22963594832064517190308341869, 5.52333952273113681477491645753, 5.87446975762017910099718519072, 5.99714360581013475599354408284, 6.04001871966838415054976521030, 6.10198838779237286729808582675, 6.65058792096698055138610119179
