| L(s) = 1 | − 4·2-s + 8·4-s − 4·7-s − 12·8-s − 16·11-s + 32·13-s + 16·14-s + 15·16-s − 40·17-s + 64·22-s + 56·23-s − 128·26-s − 32·28-s − 16·31-s − 40·32-s + 160·34-s − 64·37-s + 56·41-s + 8·43-s − 128·44-s − 224·46-s + 128·47-s + 8·49-s + 256·52-s + 56·53-s + 48·56-s + 200·61-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·4-s − 4/7·7-s − 3/2·8-s − 1.45·11-s + 2.46·13-s + 8/7·14-s + 0.937·16-s − 2.35·17-s + 2.90·22-s + 2.43·23-s − 4.92·26-s − 8/7·28-s − 0.516·31-s − 5/4·32-s + 4.70·34-s − 1.72·37-s + 1.36·41-s + 8/43·43-s − 2.90·44-s − 4.86·46-s + 2.72·47-s + 8/49·49-s + 4.92·52-s + 1.05·53-s + 6/7·56-s + 3.27·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8077096931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8077096931\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + p^{3} T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{4} T^{5} + p^{7} T^{6} + p^{8} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 156 T^{3} + 2942 T^{4} + 156 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 204 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 9120 T^{3} + 148994 T^{4} - 9120 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 15240 T^{3} + 281858 T^{4} + 15240 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 940 T^{2} + 450438 T^{4} - 940 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} - 50904 T^{3} + 1508162 T^{4} - 50904 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2128 T^{2} + 2165634 T^{4} - 2128 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 1722 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 64 T + 2048 T^{2} + 58176 T^{3} + 1440962 T^{4} + 58176 p^{2} T^{5} + 2048 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 28 T + 3342 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 5256 T^{3} - 557566 T^{4} - 5256 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 128 T + 8192 T^{2} - 506496 T^{3} + 28260194 T^{4} - 506496 p^{2} T^{5} + 8192 p^{4} T^{6} - 128 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} - 155064 T^{3} + 15333122 T^{4} - 155064 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 200 T^{2} - 5646222 T^{4} + 200 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 100 T + 7998 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 200 T + 20000 T^{2} - 1888200 T^{3} + 153742658 T^{4} - 1888200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 76 T + 2888 T^{2} - 65436 T^{3} - 36833458 T^{4} - 65436 p^{2} T^{5} + 2888 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 11882 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 101328 T^{3} + 79904642 T^{4} + 101328 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16060 T^{2} + 188845638 T^{4} - 16060 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 173820 T^{3} + 150551438 T^{4} - 173820 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−8.857511799366143256330519183974, −8.486885153823864915675400631952, −8.424693531100273962988957140831, −8.203311516710162172943877441495, −7.52257805448851540024171845167, −7.38081347429198476107736836873, −7.37966063108100749050732323340, −6.87122651976291343758957813108, −6.53357725412856894096610605873, −6.48374007198794133654159174354, −6.29239232322289397479535755583, −5.49611424865342524572353321773, −5.46132822901386991078901940701, −5.32004110061173010171602110949, −4.86716117192925319806462745489, −4.32402725310536356477860026764, −3.81519427864255003727833129861, −3.59342923529175953495772701220, −3.55654715797918749650014039214, −2.56426142020865086970385653538, −2.36140027017480890621960571235, −2.31519511012120369943232164673, −1.37564493382491436996024888761, −0.73356309336395783174297990099, −0.55708006345420739640671132517,
0.55708006345420739640671132517, 0.73356309336395783174297990099, 1.37564493382491436996024888761, 2.31519511012120369943232164673, 2.36140027017480890621960571235, 2.56426142020865086970385653538, 3.55654715797918749650014039214, 3.59342923529175953495772701220, 3.81519427864255003727833129861, 4.32402725310536356477860026764, 4.86716117192925319806462745489, 5.32004110061173010171602110949, 5.46132822901386991078901940701, 5.49611424865342524572353321773, 6.29239232322289397479535755583, 6.48374007198794133654159174354, 6.53357725412856894096610605873, 6.87122651976291343758957813108, 7.37966063108100749050732323340, 7.38081347429198476107736836873, 7.52257805448851540024171845167, 8.203311516710162172943877441495, 8.424693531100273962988957140831, 8.486885153823864915675400631952, 8.857511799366143256330519183974
