L(s) = 1 | − 2-s − 4-s + 3·8-s + 9-s + 3·11-s − 16-s − 4·17-s − 18-s + 8·19-s − 3·22-s + 2·25-s − 5·32-s + 4·34-s − 36-s − 8·38-s − 12·41-s + 16·43-s − 3·44-s − 6·49-s − 2·50-s − 16·59-s + 7·64-s − 8·67-s + 4·68-s + 3·72-s − 20·73-s − 8·76-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 1/3·9-s + 0.904·11-s − 1/4·16-s − 0.970·17-s − 0.235·18-s + 1.83·19-s − 0.639·22-s + 2/5·25-s − 0.883·32-s + 0.685·34-s − 1/6·36-s − 1.29·38-s − 1.87·41-s + 2.43·43-s − 0.452·44-s − 6/7·49-s − 0.282·50-s − 2.08·59-s + 7/8·64-s − 0.977·67-s + 0.485·68-s + 0.353·72-s − 2.34·73-s − 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6000926798\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6000926798\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.93439334804382919546995949213, −11.34497384419177131756738933013, −10.76780466456946714024310733355, −10.11672963358121359601593885655, −9.599815135362881295639350509225, −9.001815261298361697894118677486, −8.762989465694654662219980133086, −7.74426207559655217519353396530, −7.36925836009317348052168501648, −6.62619420165972560196427270392, −5.74469881629040321086611789900, −4.82090950308367352918122060264, −4.22600620193989736003151376656, −3.17606698937092513279428625412, −1.45988613491069469905927812537,
1.45988613491069469905927812537, 3.17606698937092513279428625412, 4.22600620193989736003151376656, 4.82090950308367352918122060264, 5.74469881629040321086611789900, 6.62619420165972560196427270392, 7.36925836009317348052168501648, 7.74426207559655217519353396530, 8.762989465694654662219980133086, 9.001815261298361697894118677486, 9.599815135362881295639350509225, 10.11672963358121359601593885655, 10.76780466456946714024310733355, 11.34497384419177131756738933013, 11.93439334804382919546995949213