| L(s) = 1 | + 2·5-s + 4·11-s + 4·19-s − 25-s + 4·29-s + 12·31-s + 20·41-s − 49-s + 8·55-s + 4·61-s + 28·71-s + 8·79-s − 12·89-s + 8·95-s − 12·101-s + 12·109-s − 10·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 24·155-s + 157-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 0.917·19-s − 1/5·25-s + 0.742·29-s + 2.15·31-s + 3.12·41-s − 1/7·49-s + 1.07·55-s + 0.512·61-s + 3.32·71-s + 0.900·79-s − 1.27·89-s + 0.820·95-s − 1.19·101-s + 1.14·109-s − 0.909·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 1.92·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.486244488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.486244488\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.295129513648703418563728464478, −8.146996694777704946128525018144, −7.56493494561611626219427935582, −7.54626143581179443777962769437, −6.72032754551153958501931044496, −6.55009333232478299097879497822, −6.41333172168074011557769800479, −5.93448955554269360501734963314, −5.44214532972921304361593836805, −5.32135659856487585816291865948, −4.65504719002921205025060855342, −4.39505619807262416827813399651, −3.89628260787520350511351381059, −3.65484075242296012289333356383, −2.83111565149064657528697172348, −2.72909928330320160194659919123, −2.18139691633490299136085795014, −1.62483642983660553365881613933, −0.881645498449283600742640052979, −0.867766105126444131128988026296,
0.867766105126444131128988026296, 0.881645498449283600742640052979, 1.62483642983660553365881613933, 2.18139691633490299136085795014, 2.72909928330320160194659919123, 2.83111565149064657528697172348, 3.65484075242296012289333356383, 3.89628260787520350511351381059, 4.39505619807262416827813399651, 4.65504719002921205025060855342, 5.32135659856487585816291865948, 5.44214532972921304361593836805, 5.93448955554269360501734963314, 6.41333172168074011557769800479, 6.55009333232478299097879497822, 6.72032754551153958501931044496, 7.54626143581179443777962769437, 7.56493494561611626219427935582, 8.146996694777704946128525018144, 8.295129513648703418563728464478
