| L(s) = 1 | + 2·7-s − 6·9-s − 4·17-s − 16·23-s − 2·25-s − 16·31-s + 12·41-s − 16·47-s + 3·49-s − 12·63-s − 16·71-s − 12·73-s + 16·79-s + 27·81-s − 12·89-s + 28·97-s − 12·113-s − 8·119-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 2·9-s − 0.970·17-s − 3.33·23-s − 2/5·25-s − 2.87·31-s + 1.87·41-s − 2.33·47-s + 3/7·49-s − 1.51·63-s − 1.89·71-s − 1.40·73-s + 1.80·79-s + 3·81-s − 1.27·89-s + 2.84·97-s − 1.12·113-s − 0.733·119-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 + 1.94·153-s + 0.0798·157-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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.900881667443540017079258183694, −8.799236922178969684769191168121, −8.188591048811035804703523217655, −8.045245190638781218099304120346, −7.54412029242978308809836179681, −7.36168047730829337949496504851, −6.45630596817959378454372774453, −6.22902841608997674978941157276, −5.71630079190482484194003814921, −5.70240799747352037019372316548, −5.00149918183437216196852079022, −4.60840781381815747813842834000, −3.86103026457261742937987830198, −3.80060917499115035392136701315, −3.08307298683603365154117235205, −2.34052236065239448543253357721, −2.12698556059685586514534277950, −1.54194435066604884302410940401, 0, 0,
1.54194435066604884302410940401, 2.12698556059685586514534277950, 2.34052236065239448543253357721, 3.08307298683603365154117235205, 3.80060917499115035392136701315, 3.86103026457261742937987830198, 4.60840781381815747813842834000, 5.00149918183437216196852079022, 5.70240799747352037019372316548, 5.71630079190482484194003814921, 6.22902841608997674978941157276, 6.45630596817959378454372774453, 7.36168047730829337949496504851, 7.54412029242978308809836179681, 8.045245190638781218099304120346, 8.188591048811035804703523217655, 8.799236922178969684769191168121, 8.900881667443540017079258183694
