| L(s) = 1 | − 8·9-s − 16·31-s − 16·49-s − 48·71-s − 16·79-s + 30·81-s + 24·89-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 8/3·9-s − 2.87·31-s − 2.28·49-s − 5.69·71-s − 1.80·79-s + 10/3·81-s + 2.54·89-s − 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.06850419758579533786330567747, −5.80197538562869542651259440784, −5.60378394911486187080527239048, −5.58492013741153305878398206917, −5.29292415720471260892670164542, −5.10275587523838363260524840969, −4.94675267202600344744495208992, −4.65843070497550566500313198064, −4.65630692974678314939297179735, −4.38098498604976843449932158069, −3.98079298463357834445392266369, −3.92011617226224657915047754052, −3.64191608388784036272786074950, −3.38857621263984253441279364423, −3.22117862106920248366625643546, −3.14465527979256879059952909959, −3.06960392737345797667587767594, −2.47780175999578407289852358161, −2.41056262494820274165743846940, −2.37164588593200543476549377681, −2.13707837471348945298554068463, −1.56420942260778860236921431298, −1.45306303919754013875753306283, −1.26078263625624750153622647483, −1.00212386075222346830219885388, 0, 0, 0, 0,
1.00212386075222346830219885388, 1.26078263625624750153622647483, 1.45306303919754013875753306283, 1.56420942260778860236921431298, 2.13707837471348945298554068463, 2.37164588593200543476549377681, 2.41056262494820274165743846940, 2.47780175999578407289852358161, 3.06960392737345797667587767594, 3.14465527979256879059952909959, 3.22117862106920248366625643546, 3.38857621263984253441279364423, 3.64191608388784036272786074950, 3.92011617226224657915047754052, 3.98079298463357834445392266369, 4.38098498604976843449932158069, 4.65630692974678314939297179735, 4.65843070497550566500313198064, 4.94675267202600344744495208992, 5.10275587523838363260524840969, 5.29292415720471260892670164542, 5.58492013741153305878398206917, 5.60378394911486187080527239048, 5.80197538562869542651259440784, 6.06850419758579533786330567747
