L(s) = 1 | − 2·7-s − 4·9-s − 12·17-s + 12·23-s − 8·25-s − 8·31-s − 12·41-s + 3·49-s + 8·63-s − 4·73-s + 16·79-s + 7·81-s − 12·89-s − 20·97-s + 8·103-s − 24·113-s + 24·119-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s − 24·161-s + ⋯ |
L(s) = 1 | − 0.755·7-s − 4/3·9-s − 2.91·17-s + 2.50·23-s − 8/5·25-s − 1.43·31-s − 1.87·41-s + 3/7·49-s + 1.00·63-s − 0.468·73-s + 1.80·79-s + 7/9·81-s − 1.27·89-s − 2.03·97-s + 0.788·103-s − 2.25·113-s + 2.20·119-s − 1.27·121-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 + 3.88·153-s + 0.0798·157-s − 1.89·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.940924759521778181505379343724, −8.917248519513164165984406257004, −8.283228406207846638253734639734, −8.164877637411407172200609504582, −7.28608270441302365439431710489, −7.08915894283415908958162536442, −6.63180254507497902809563536380, −6.46212586418590935164233249435, −5.76597064740174175738918312023, −5.56776808203097827554404418361, −4.83844996195026524900711874308, −4.78778939165836108718564571290, −3.86964436952725826027560565534, −3.67960519229877972775828601242, −3.02895939590336338139094924453, −2.53469516521731775666477050470, −2.18371094795830213388608812376, −1.37582359136721476244828693567, 0, 0,
1.37582359136721476244828693567, 2.18371094795830213388608812376, 2.53469516521731775666477050470, 3.02895939590336338139094924453, 3.67960519229877972775828601242, 3.86964436952725826027560565534, 4.78778939165836108718564571290, 4.83844996195026524900711874308, 5.56776808203097827554404418361, 5.76597064740174175738918312023, 6.46212586418590935164233249435, 6.63180254507497902809563536380, 7.08915894283415908958162536442, 7.28608270441302365439431710489, 8.164877637411407172200609504582, 8.283228406207846638253734639734, 8.917248519513164165984406257004, 8.940924759521778181505379343724