| L(s) = 1 | − 6·3-s + 6·5-s + 8·7-s + 16·9-s − 12·11-s + 6·13-s − 36·15-s − 12·17-s + 2·19-s − 48·21-s − 6·23-s + 18·25-s − 24·27-s − 6·29-s + 8·31-s + 72·33-s + 48·35-s + 8·37-s − 36·39-s − 12·41-s + 4·43-s + 96·45-s + 24·47-s + 44·49-s + 72·51-s + 24·53-s − 72·55-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 2.68·5-s + 3.02·7-s + 16/3·9-s − 3.61·11-s + 1.66·13-s − 9.29·15-s − 2.91·17-s + 0.458·19-s − 10.4·21-s − 1.25·23-s + 18/5·25-s − 4.61·27-s − 1.11·29-s + 1.43·31-s + 12.5·33-s + 8.11·35-s + 1.31·37-s − 5.76·39-s − 1.87·41-s + 0.609·43-s + 14.3·45-s + 3.50·47-s + 44/7·49-s + 10.0·51-s + 3.29·53-s − 9.70·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.152564707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.152564707\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 p T + 20 T^{2} + 16 p T^{3} + 91 T^{4} + 16 p^{2} T^{5} + 20 p^{2} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 8 T + 20 T^{2} + 12 T^{3} - 145 T^{4} + 12 p T^{5} + 20 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 288 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 5 p T^{2} + 444 T^{3} + 1896 T^{4} + 444 p T^{5} + 5 p^{3} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 264 T^{3} + 2174 T^{4} - 264 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 117 T^{2} + 840 T^{3} + 5804 T^{4} + 840 p T^{5} + 117 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2712 T^{3} + 21182 T^{4} - 2712 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 139 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 36 T^{2} - 12 p T^{3} - 9001 T^{4} - 12 p^{2} T^{5} + 36 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 95 T^{2} + 5304 T^{4} - 95 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 10 T + 194 T^{2} - 1644 T^{3} + 18239 T^{4} - 1644 p T^{5} + 194 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} - 36 T^{3} - 4153 T^{4} - 36 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 14 T + 98 T^{2} + 1176 T^{3} + 13991 T^{4} + 1176 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 12870 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 564 T^{3} + 3122 T^{4} + 564 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 18 T + 162 T^{2} - 972 T^{3} + 4607 T^{4} - 972 p T^{5} + 162 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 26 T + 2 p T^{2} + 1332 T^{3} - 32593 T^{4} + 1332 p T^{5} + 2 p^{3} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.36039100459451827030183113297, −7.01114795513029696137610828163, −6.74649012842567371443288662366, −6.28792554110948242848040496863, −6.09305563543480897186860096597, −6.00650846145920164572244645303, −5.93562871194940635801598101801, −5.75974631652799478148637042073, −5.50973482323966908056885013825, −5.32250500447841829932095173491, −4.99879187951165816534608027134, −4.96866723654543807378392220498, −4.86405898065164734075100601748, −4.54107629869485486917867626128, −4.21922874103801157219867621943, −3.95153074090327286059396727939, −3.53436304763043971305971134997, −2.56296925648340434243643081461, −2.49311733868751128281371914291, −2.39191378306723175061536677328, −1.98993652277652155637907004600, −1.94830438493260294608273274510, −1.20535291398892866975302546881, −0.906688163752870335751279372566, −0.40660087288661687601838351112,
0.40660087288661687601838351112, 0.906688163752870335751279372566, 1.20535291398892866975302546881, 1.94830438493260294608273274510, 1.98993652277652155637907004600, 2.39191378306723175061536677328, 2.49311733868751128281371914291, 2.56296925648340434243643081461, 3.53436304763043971305971134997, 3.95153074090327286059396727939, 4.21922874103801157219867621943, 4.54107629869485486917867626128, 4.86405898065164734075100601748, 4.96866723654543807378392220498, 4.99879187951165816534608027134, 5.32250500447841829932095173491, 5.50973482323966908056885013825, 5.75974631652799478148637042073, 5.93562871194940635801598101801, 6.00650846145920164572244645303, 6.09305563543480897186860096597, 6.28792554110948242848040496863, 6.74649012842567371443288662366, 7.01114795513029696137610828163, 7.36039100459451827030183113297
