| L(s) = 1 | − 2·4-s + 4·7-s + 4·16-s + 2·25-s − 8·28-s − 20·31-s − 2·49-s − 8·64-s + 28·73-s − 20·79-s + 4·97-s − 4·100-s + 28·103-s + 16·112-s − 10·121-s + 40·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4-s + 1.51·7-s + 16-s + 2/5·25-s − 1.51·28-s − 3.59·31-s − 2/7·49-s − 64-s + 3.27·73-s − 2.25·79-s + 0.406·97-s − 2/5·100-s + 2.75·103-s + 1.51·112-s − 0.909·121-s + 3.59·124-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 + 2·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7839001763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7839001763\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−14.67775387191730282028398531410, −14.28909176397700135196412256125, −14.17434646576563573812079270815, −13.18532026363482569767025347565, −12.84210040565858777461327977235, −12.35718091867383862347936968396, −11.40237991051490769499946060660, −11.19071334099707376619912511674, −10.51537962851961349002635536516, −9.831838475826753763234022542131, −8.920752578114195302722929551822, −8.916367444552218392773030973974, −7.80946111775523970186167491896, −7.70320398882244853493446925815, −6.63376132110711059298421719352, −5.44701643907424513444627594829, −5.19534403732119005925496553562, −4.31228352047829502616019460801, −3.49994922448227040508054853507, −1.78250116892438199199236003766,
1.78250116892438199199236003766, 3.49994922448227040508054853507, 4.31228352047829502616019460801, 5.19534403732119005925496553562, 5.44701643907424513444627594829, 6.63376132110711059298421719352, 7.70320398882244853493446925815, 7.80946111775523970186167491896, 8.916367444552218392773030973974, 8.920752578114195302722929551822, 9.831838475826753763234022542131, 10.51537962851961349002635536516, 11.19071334099707376619912511674, 11.40237991051490769499946060660, 12.35718091867383862347936968396, 12.84210040565858777461327977235, 13.18532026363482569767025347565, 14.17434646576563573812079270815, 14.28909176397700135196412256125, 14.67775387191730282028398531410
