| L(s) = 1 | − 6·3-s + 6·7-s + 21·9-s − 6·11-s − 6·13-s − 12·17-s − 36·21-s − 6·23-s − 54·27-s + 12·29-s − 2·31-s + 36·33-s + 12·37-s + 36·39-s + 12·41-s + 6·43-s − 24·47-s + 27·49-s + 72·51-s + 12·53-s − 12·59-s − 4·61-s + 126·63-s + 30·67-s + 36·69-s + 24·73-s − 36·77-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 2.26·7-s + 7·9-s − 1.80·11-s − 1.66·13-s − 2.91·17-s − 7.85·21-s − 1.25·23-s − 10.3·27-s + 2.22·29-s − 0.359·31-s + 6.26·33-s + 1.97·37-s + 5.76·39-s + 1.87·41-s + 0.914·43-s − 3.50·47-s + 27/7·49-s + 10.0·51-s + 1.64·53-s − 1.56·59-s − 0.512·61-s + 15.8·63-s + 3.66·67-s + 4.33·69-s + 2.80·73-s − 4.10·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4801043747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4801043747\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 6 T + 9 T^{2} + 6 p T^{3} - 208 T^{4} + 6 p^{2} T^{5} + 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 28 T^{2} + 96 T^{3} + 267 T^{4} + 96 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 + 6 T + 18 T^{2} + 36 T^{3} + 23 T^{4} + 36 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 714 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 246 T^{3} + 1208 T^{4} + 246 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 12 T + 53 T^{2} - 396 T^{3} + 3264 T^{4} - 396 p T^{5} + 53 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 2 T - 32 T^{2} - 52 T^{3} + 211 T^{4} - 52 p T^{5} - 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 372 T^{3} + 1886 T^{4} - 372 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 133 T^{2} - 1020 T^{3} + 7512 T^{4} - 1020 p T^{5} + 133 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} + 408 T^{3} - 3073 T^{4} + 408 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 24 T + 369 T^{2} + 3792 T^{3} + 30092 T^{4} + 3792 p T^{5} + 369 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 204 T^{3} - 718 T^{4} - 204 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 2 T^{2} + 288 T^{3} + 8187 T^{4} + 288 p T^{5} + 2 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T - 83 T^{2} - 92 T^{3} + 5104 T^{4} - 92 p T^{5} - 83 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 30 T + 549 T^{2} - 6774 T^{3} + 63824 T^{4} - 6774 p T^{5} + 549 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 10746 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2904 T^{3} + 26978 T^{4} - 2904 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 + 154 T^{2} + 17475 T^{4} + 154 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 81 T^{2} + 1008 T^{3} + 2804 T^{4} + 1008 p T^{5} + 81 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 139 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 24 T + 180 T^{2} + 1620 T^{3} - 36289 T^{4} + 1620 p T^{5} + 180 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.11977367676253946305443093167, −7.02127004954535407433022962917, −6.67319011069847259915738611738, −6.52301352726922353596929558623, −6.27485206249615668559538646102, −6.02730868526862113226654331350, −5.94711399639864225393016763499, −5.61690439821922971442266843482, −5.23035472378565361562715035880, −5.06432158844461556894456721093, −4.92930649591799834474502932892, −4.81592091391900501737428390463, −4.63290938053960756750500096117, −4.58051507615867807484383962929, −4.01358707857413766578542507761, −4.00803023373045572974715814092, −3.66164441229851467818749920290, −2.56671857718475366822066740581, −2.54463913953446417553051502020, −2.44579224678175686341237689175, −2.12299797030015796566480533458, −1.60473220361100890060172390638, −1.18092134555463192182178331659, −0.62911987502932019083612547727, −0.35671969289575346207616796072,
0.35671969289575346207616796072, 0.62911987502932019083612547727, 1.18092134555463192182178331659, 1.60473220361100890060172390638, 2.12299797030015796566480533458, 2.44579224678175686341237689175, 2.54463913953446417553051502020, 2.56671857718475366822066740581, 3.66164441229851467818749920290, 4.00803023373045572974715814092, 4.01358707857413766578542507761, 4.58051507615867807484383962929, 4.63290938053960756750500096117, 4.81592091391900501737428390463, 4.92930649591799834474502932892, 5.06432158844461556894456721093, 5.23035472378565361562715035880, 5.61690439821922971442266843482, 5.94711399639864225393016763499, 6.02730868526862113226654331350, 6.27485206249615668559538646102, 6.52301352726922353596929558623, 6.67319011069847259915738611738, 7.02127004954535407433022962917, 7.11977367676253946305443093167
