| L(s) = 1 | + 2·3-s + 2·9-s − 12·11-s + 8·23-s + 8·25-s + 6·27-s − 24·33-s − 16·37-s + 32·47-s + 16·49-s − 4·59-s + 16·61-s + 16·69-s + 8·71-s − 8·73-s + 16·75-s + 11·81-s − 20·83-s − 16·97-s − 24·99-s + 12·107-s − 16·109-s − 32·111-s + 56·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2/3·9-s − 3.61·11-s + 1.66·23-s + 8/5·25-s + 1.15·27-s − 4.17·33-s − 2.63·37-s + 4.66·47-s + 16/7·49-s − 0.520·59-s + 2.04·61-s + 1.92·69-s + 0.949·71-s − 0.936·73-s + 1.84·75-s + 11/9·81-s − 2.19·83-s − 1.62·97-s − 2.41·99-s + 1.16·107-s − 1.53·109-s − 3.03·111-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.488575444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488575444\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 46 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4$ | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 1118 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 718 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 6334 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 15598 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7758 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 10 T + 186 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.218054529458063796990288542526, −8.095712444197151852803827793526, −7.81154220456290406141470382148, −7.31788822670226719981440106942, −7.31167834096556364237367427665, −7.04115419664090146379155592869, −6.87775482818256847204336463769, −6.87608787923098798869876049769, −5.92109346824573554556271882355, −5.90825067597875359038969952359, −5.58033037672935558922287635504, −5.30617010774444546975566291979, −5.06366884368233726996521205517, −5.01193824464423276237475718191, −4.59214357569285728443681311873, −4.08716589593396748958705805894, −3.98058799524650802644509326142, −3.49651607573513232125593208464, −2.99189521466023003254220494210, −2.81637949391504147846896050728, −2.69981003140452455874462258290, −2.38816110549454468176908182544, −2.04424415332624413801861504578, −1.18538501525557478656219182913, −0.61237961889001587365307450298,
0.61237961889001587365307450298, 1.18538501525557478656219182913, 2.04424415332624413801861504578, 2.38816110549454468176908182544, 2.69981003140452455874462258290, 2.81637949391504147846896050728, 2.99189521466023003254220494210, 3.49651607573513232125593208464, 3.98058799524650802644509326142, 4.08716589593396748958705805894, 4.59214357569285728443681311873, 5.01193824464423276237475718191, 5.06366884368233726996521205517, 5.30617010774444546975566291979, 5.58033037672935558922287635504, 5.90825067597875359038969952359, 5.92109346824573554556271882355, 6.87608787923098798869876049769, 6.87775482818256847204336463769, 7.04115419664090146379155592869, 7.31167834096556364237367427665, 7.31788822670226719981440106942, 7.81154220456290406141470382148, 8.095712444197151852803827793526, 8.218054529458063796990288542526
