| L(s) = 1 | + 4·2-s + 10·4-s − 4·5-s − 2·7-s + 20·8-s − 16·10-s − 10·11-s − 2·13-s − 8·14-s + 35·16-s + 6·17-s + 8·19-s − 40·20-s − 40·22-s − 6·23-s + 10·25-s − 8·26-s − 20·28-s − 14·29-s + 56·32-s + 24·34-s + 8·35-s − 10·37-s + 32·38-s − 80·40-s − 2·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 1.78·5-s − 0.755·7-s + 7.07·8-s − 5.05·10-s − 3.01·11-s − 0.554·13-s − 2.13·14-s + 35/4·16-s + 1.45·17-s + 1.83·19-s − 8.94·20-s − 8.52·22-s − 1.25·23-s + 2·25-s − 1.56·26-s − 3.77·28-s − 2.59·29-s + 9.89·32-s + 4.11·34-s + 1.35·35-s − 1.64·37-s + 5.19·38-s − 12.6·40-s − 0.312·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 67^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 67 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 + 2 T + 8 T^{2} + 10 T^{3} + 46 T^{4} + 10 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 10 T + 58 T^{2} + 2 p^{2} T^{3} + 874 T^{4} + 2 p^{3} T^{5} + 58 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 2 T + 20 T^{2} + 54 T^{3} + 30 p T^{4} + 54 p T^{5} + 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 6 T + 26 T^{2} - 154 T^{3} + 914 T^{4} - 154 p T^{5} + 26 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 8 T + 64 T^{2} - 312 T^{3} + 1646 T^{4} - 312 p T^{5} + 64 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 6 T + 56 T^{2} + 158 T^{3} + 1262 T^{4} + 158 p T^{5} + 56 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 14 T + 120 T^{2} + 770 T^{3} + 4478 T^{4} + 770 p T^{5} + 120 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 26 T^{2} - 12 T^{3} + 1954 T^{4} - 12 p T^{5} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 10 T + 110 T^{2} + 582 T^{3} + 4434 T^{4} + 582 p T^{5} + 110 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 2 T + 142 T^{2} + 190 T^{3} + 8290 T^{4} + 190 p T^{5} + 142 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 4 T + 80 T^{2} + 196 T^{3} + 4206 T^{4} + 196 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 18 T + 288 T^{2} + 2698 T^{3} + 22622 T^{4} + 2698 p T^{5} + 288 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 4 T + 40 T^{2} - 332 T^{3} + 5950 T^{4} - 332 p T^{5} + 40 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 28 T + 480 T^{2} + 5516 T^{3} + 48942 T^{4} + 5516 p T^{5} + 480 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 28 T + 466 T^{2} + 5400 T^{3} + 47978 T^{4} + 5400 p T^{5} + 466 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 6 T + 156 T^{2} + 1078 T^{3} + 12086 T^{4} + 1078 p T^{5} + 156 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 12 T - 4 T^{2} + 628 T^{3} + 15846 T^{4} + 628 p T^{5} - 4 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 4 T + 202 T^{2} + 480 T^{3} + 19394 T^{4} + 480 p T^{5} + 202 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 14 T + 306 T^{2} + 3150 T^{3} + 36890 T^{4} + 3150 p T^{5} + 306 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 4 T + 268 T^{2} + 1148 T^{3} + 32518 T^{4} + 1148 p T^{5} + 268 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 8 T + 200 T^{2} + 2120 T^{3} + 21582 T^{4} + 2120 p T^{5} + 200 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.98724717411462830595007917113, −5.88537713465234367933364571185, −5.48743735310315672612691885485, −5.37622448810879362641348870833, −5.22628665886938795346323303512, −5.05936631499114522845529866599, −4.93307902486004658738882140560, −4.88896781510742588311146271605, −4.60523724137589320429139376748, −4.26654438182385578270995232431, −4.08851621588946779067740350430, −4.08432432135515941184848032019, −3.66707200823923010177571805817, −3.42225333196731920798182205432, −3.33469157345300066222930713122, −3.32045469675287679981659229194, −3.03261479011271626939439228239, −2.83807382735504133998012297101, −2.69484860594087387916026380060, −2.59399703226325534454288461765, −2.21693161842788102647846750022, −1.70664271087700849604238305184, −1.47111625580531283811875517877, −1.43444921983442575613122225837, −1.32438110538439658604396735820, 0, 0, 0, 0,
1.32438110538439658604396735820, 1.43444921983442575613122225837, 1.47111625580531283811875517877, 1.70664271087700849604238305184, 2.21693161842788102647846750022, 2.59399703226325534454288461765, 2.69484860594087387916026380060, 2.83807382735504133998012297101, 3.03261479011271626939439228239, 3.32045469675287679981659229194, 3.33469157345300066222930713122, 3.42225333196731920798182205432, 3.66707200823923010177571805817, 4.08432432135515941184848032019, 4.08851621588946779067740350430, 4.26654438182385578270995232431, 4.60523724137589320429139376748, 4.88896781510742588311146271605, 4.93307902486004658738882140560, 5.05936631499114522845529866599, 5.22628665886938795346323303512, 5.37622448810879362641348870833, 5.48743735310315672612691885485, 5.88537713465234367933364571185, 5.98724717411462830595007917113
