| L(s) = 1 | + 6·3-s − 8·5-s + 6·7-s + 21·9-s + 10·11-s − 10·13-s − 48·15-s − 16·17-s + 6·19-s + 36·21-s − 6·23-s + 44·25-s + 54·27-s − 6·29-s + 8·31-s + 60·33-s − 48·35-s + 12·37-s − 60·39-s + 16·43-s − 168·45-s + 2·47-s + 8·49-s − 96·51-s − 16·53-s − 80·55-s + 36·57-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 3.57·5-s + 2.26·7-s + 7·9-s + 3.01·11-s − 2.77·13-s − 12.3·15-s − 3.88·17-s + 1.37·19-s + 7.85·21-s − 1.25·23-s + 44/5·25-s + 10.3·27-s − 1.11·29-s + 1.43·31-s + 10.4·33-s − 8.11·35-s + 1.97·37-s − 9.60·39-s + 2.43·43-s − 25.0·45-s + 0.291·47-s + 8/7·49-s − 13.4·51-s − 2.19·53-s − 10.7·55-s + 4.76·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.733023320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.733023320\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 8 T + 4 p T^{2} - 4 T^{3} - 89 T^{4} - 4 p T^{5} + 4 p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 6 T + 4 p T^{2} - 96 T^{3} + 291 T^{4} - 96 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 10 T + 41 T^{2} - 82 T^{3} + 136 T^{4} - 82 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 132 T^{3} + 959 T^{4} - 132 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 52 T^{2} + 240 T^{3} + 1347 T^{4} + 240 p T^{5} + 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 6 T + 18 T^{2} + 36 T^{3} - 457 T^{4} + 36 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T - 2 T^{2} - 32 T^{3} + 1411 T^{4} - 32 p T^{5} - 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 588 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 16 T + 65 T^{2} + 624 T^{3} - 8092 T^{4} + 624 p T^{5} + 65 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 148 T^{3} - 1997 T^{4} + 148 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 976 T^{3} + 7378 T^{4} + 976 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T + 45 T^{2} - 594 T^{3} - 3376 T^{4} - 594 p T^{5} + 45 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 180 T^{2} + 1596 T^{3} + 15143 T^{4} + 1596 p T^{5} + 180 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 113 T^{2} + 384 T^{3} - 172 T^{4} + 384 p T^{5} + 113 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 2 T + 2 T^{2} + 328 T^{3} - 7217 T^{4} + 328 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 20 T + 109 T^{2} + 20 p T^{3} + 376 p T^{4} + 20 p^{2} T^{5} + 109 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.87717361387222809149298613794, −7.47500259637307731378188261310, −7.45522003395365809876340972527, −7.26120610036209112453457874834, −7.20392190230465880518958012204, −6.91859964322026948445517198573, −6.44067288560019112536254488185, −6.26041863570242261577086910155, −6.10517468183462305665122457970, −5.04513515440582730632390454204, −4.80262701270519440538109749946, −4.58169251470589773119412884980, −4.52894018041374925835526999335, −4.41212321747056481783418730999, −4.22426791487876113833159390421, −4.07698259689205053550646436199, −3.49327055423267326069165901655, −3.46159142506378581515616693618, −3.01140495584972192352855143522, −2.79285429412326055230635552274, −2.40603293811004784736397263174, −1.93184635932870062166391627912, −1.91493704273533030447488103580, −1.26848366388194224536007203471, −0.64718781797248432170085297969,
0.64718781797248432170085297969, 1.26848366388194224536007203471, 1.91493704273533030447488103580, 1.93184635932870062166391627912, 2.40603293811004784736397263174, 2.79285429412326055230635552274, 3.01140495584972192352855143522, 3.46159142506378581515616693618, 3.49327055423267326069165901655, 4.07698259689205053550646436199, 4.22426791487876113833159390421, 4.41212321747056481783418730999, 4.52894018041374925835526999335, 4.58169251470589773119412884980, 4.80262701270519440538109749946, 5.04513515440582730632390454204, 6.10517468183462305665122457970, 6.26041863570242261577086910155, 6.44067288560019112536254488185, 6.91859964322026948445517198573, 7.20392190230465880518958012204, 7.26120610036209112453457874834, 7.45522003395365809876340972527, 7.47500259637307731378188261310, 7.87717361387222809149298613794
