| L(s) = 1 | − 4-s + 2·9-s + 6·11-s + 16-s − 4·19-s − 6·29-s + 10·31-s − 2·36-s + 6·41-s − 6·44-s + 13·49-s − 2·61-s − 64-s + 12·71-s + 4·76-s − 16·79-s − 5·81-s + 12·89-s + 12·99-s − 12·101-s − 22·109-s + 6·116-s + 5·121-s − 10·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s + 1.80·11-s + 1/4·16-s − 0.917·19-s − 1.11·29-s + 1.79·31-s − 1/3·36-s + 0.937·41-s − 0.904·44-s + 13/7·49-s − 0.256·61-s − 1/8·64-s + 1.42·71-s + 0.458·76-s − 1.80·79-s − 5/9·81-s + 1.27·89-s + 1.20·99-s − 1.19·101-s − 2.10·109-s + 0.557·116-s + 5/11·121-s − 0.898·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.492450406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.492450406\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−9.469795911187362942598550550258, −8.950746532704195704092773024545, −8.899177216290367638252314334019, −8.129848952162743451971820262235, −8.109686740664073178919541779159, −7.42751653773487178162056300692, −6.88678194097256094925194069278, −6.86002888593224999857280811535, −6.15597997467289713825454436392, −6.01244020853163183918690084216, −5.39845382891670283639741923816, −4.90775290295400081722568089157, −4.23207593060098564990426900551, −4.13073344291865515346216601470, −3.89246028242719574943863524280, −3.10192255222434719448342467617, −2.54604595404544582857000315103, −1.84889183125781260900045379826, −1.29159593380292501720843578493, −0.65293521367922629127572194155,
0.65293521367922629127572194155, 1.29159593380292501720843578493, 1.84889183125781260900045379826, 2.54604595404544582857000315103, 3.10192255222434719448342467617, 3.89246028242719574943863524280, 4.13073344291865515346216601470, 4.23207593060098564990426900551, 4.90775290295400081722568089157, 5.39845382891670283639741923816, 6.01244020853163183918690084216, 6.15597997467289713825454436392, 6.86002888593224999857280811535, 6.88678194097256094925194069278, 7.42751653773487178162056300692, 8.109686740664073178919541779159, 8.129848952162743451971820262235, 8.899177216290367638252314334019, 8.950746532704195704092773024545, 9.469795911187362942598550550258
