| L(s) = 1 | + 4·3-s + 8·7-s + 8·9-s − 8·13-s − 8·17-s − 20·19-s + 32·21-s − 8·23-s + 8·27-s + 24·37-s − 32·39-s + 20·41-s − 8·43-s + 32·49-s − 32·51-s + 16·53-s − 80·57-s − 8·59-s − 16·61-s + 64·63-s + 12·67-s − 32·69-s + 8·73-s − 16·79-s − 7·81-s + 4·83-s − 64·91-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3.02·7-s + 8/3·9-s − 2.21·13-s − 1.94·17-s − 4.58·19-s + 6.98·21-s − 1.66·23-s + 1.53·27-s + 3.94·37-s − 5.12·39-s + 3.12·41-s − 1.21·43-s + 32/7·49-s − 4.48·51-s + 2.19·53-s − 10.5·57-s − 1.04·59-s − 2.04·61-s + 8.06·63-s + 1.46·67-s − 3.85·69-s + 0.936·73-s − 1.80·79-s − 7/9·81-s + 0.439·83-s − 6.70·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.283727265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.283727265\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 4 T + 8 T^{2} - 8 T^{3} + 7 T^{4} - 8 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 443 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 72 T^{3} + 146 T^{4} + 72 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 176 T^{3} + 943 T^{4} + 176 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 152 T^{3} + 706 T^{4} + 152 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 2198 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2694 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 24 T^{3} - 1582 T^{4} + 24 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 4222 T^{4} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1168 T^{3} + 10258 T^{4} - 1168 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 984 T^{3} + 13223 T^{4} - 984 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 432 T^{3} + 5471 T^{4} - 432 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 232 T^{3} + 6103 T^{4} - 232 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 210 T^{2} + 22163 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 2498 T^{4} + p^{4} T^{8} \) |
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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
−6.68440870174955719242640996180, −6.66265940986880257596001306400, −6.19050051187204496339002275983, −5.99629772393148244605698293267, −5.98067923478083920555253865189, −5.81688797234448493427090284671, −5.18418106029440650458433777534, −5.00150591070877674288942659308, −4.75482255233381013595698201532, −4.52611261655381970079971498081, −4.45151598215290668023308600529, −4.20655257577612034842059707199, −4.15163425532897595350115782768, −4.04573213808522257409008805859, −3.73558470578260884849002753413, −3.03703835134878381767086394289, −2.69923005278298236605425852143, −2.65795503160057698719055425589, −2.38024053537266549203576939131, −2.19486171536224676955764049623, −1.92612035014762841065554449312, −1.90140826912479436584353861861, −1.69960715053317090113101471159, −0.825913369565941238279379737066, −0.39219721297220349435477077797,
0.39219721297220349435477077797, 0.825913369565941238279379737066, 1.69960715053317090113101471159, 1.90140826912479436584353861861, 1.92612035014762841065554449312, 2.19486171536224676955764049623, 2.38024053537266549203576939131, 2.65795503160057698719055425589, 2.69923005278298236605425852143, 3.03703835134878381767086394289, 3.73558470578260884849002753413, 4.04573213808522257409008805859, 4.15163425532897595350115782768, 4.20655257577612034842059707199, 4.45151598215290668023308600529, 4.52611261655381970079971498081, 4.75482255233381013595698201532, 5.00150591070877674288942659308, 5.18418106029440650458433777534, 5.81688797234448493427090284671, 5.98067923478083920555253865189, 5.99629772393148244605698293267, 6.19050051187204496339002275983, 6.66265940986880257596001306400, 6.68440870174955719242640996180
