| L(s) = 1 | − 3-s − 2·5-s − 4·11-s + 4·13-s + 2·15-s + 2·17-s − 4·19-s + 8·23-s + 5·25-s + 27-s + 12·29-s + 8·31-s + 4·33-s − 6·37-s − 4·39-s + 12·41-s + 8·43-s − 2·51-s + 2·53-s + 8·55-s + 4·57-s + 4·59-s − 2·61-s − 8·65-s + 4·67-s − 8·69-s + 16·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.20·11-s + 1.10·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 25-s + 0.192·27-s + 2.22·29-s + 1.43·31-s + 0.696·33-s − 0.986·37-s − 0.640·39-s + 1.87·41-s + 1.21·43-s − 0.280·51-s + 0.274·53-s + 1.07·55-s + 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.992·65-s + 0.488·67-s − 0.963·69-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.623859576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.623859576\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 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
−10.10217233970793979369065060446, −9.619368421009300866508913914680, −9.081203540834864469157483391444, −8.508560733923636878536261819888, −8.380298627616214674784080934621, −8.116306881006753619706238728616, −7.37300450662439718491386993296, −7.22234137635056789428936445382, −6.48997705313110646632417027104, −6.34208523717648861499136733178, −5.80775818102949760271055661261, −5.20575579336004621938561904655, −4.68487282158073792164799368383, −4.66476015027554982333922112470, −3.76578752139690222452954507812, −3.42625582674721034858861047812, −2.64933603993002024835025707506, −2.44358454618033908230511650105, −0.993295613762746659183663965773, −0.76932013414470230191780125820,
0.76932013414470230191780125820, 0.993295613762746659183663965773, 2.44358454618033908230511650105, 2.64933603993002024835025707506, 3.42625582674721034858861047812, 3.76578752139690222452954507812, 4.66476015027554982333922112470, 4.68487282158073792164799368383, 5.20575579336004621938561904655, 5.80775818102949760271055661261, 6.34208523717648861499136733178, 6.48997705313110646632417027104, 7.22234137635056789428936445382, 7.37300450662439718491386993296, 8.116306881006753619706238728616, 8.380298627616214674784080934621, 8.508560733923636878536261819888, 9.081203540834864469157483391444, 9.619368421009300866508913914680, 10.10217233970793979369065060446
