| L(s) = 1 | + 2-s + 4·5-s − 8-s − 3·9-s + 4·10-s + 2·11-s + 6·13-s − 16-s + 5·17-s − 3·18-s + 7·19-s + 2·22-s + 8·23-s + 2·25-s + 6·26-s + 4·29-s + 6·31-s + 5·34-s − 2·37-s + 7·38-s − 4·40-s − 3·41-s + 43-s − 12·45-s + 8·46-s + 2·50-s − 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.78·5-s − 0.353·8-s − 9-s + 1.26·10-s + 0.603·11-s + 1.66·13-s − 1/4·16-s + 1.21·17-s − 0.707·18-s + 1.60·19-s + 0.426·22-s + 1.66·23-s + 2/5·25-s + 1.17·26-s + 0.742·29-s + 1.07·31-s + 0.857·34-s − 0.328·37-s + 1.13·38-s − 0.632·40-s − 0.468·41-s + 0.152·43-s − 1.78·45-s + 1.17·46-s + 0.282·50-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.907714229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.907714229\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 16 T + 173 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.22320619755545118655195289616, −9.976447831725577390742809063318, −9.532557728911292126627635924558, −9.062811038441739238282432200803, −8.775126213065446762474526089106, −8.465615571691745927792215072143, −7.73441976365842372781486087829, −7.38054044225898634615941577805, −6.58841792152257971985007080180, −6.18336546061230623342307495037, −6.08476828479604175178802693467, −5.53104872938569287196583548776, −5.15165993315668965696796025589, −4.85149636420972615815433211238, −3.86494639311876150628217678616, −3.46695233978356895747322902688, −2.95345304042477107874750155634, −2.51905966752056633678736527264, −1.32099734781954153468787152585, −1.24216478288915097802112651595,
1.24216478288915097802112651595, 1.32099734781954153468787152585, 2.51905966752056633678736527264, 2.95345304042477107874750155634, 3.46695233978356895747322902688, 3.86494639311876150628217678616, 4.85149636420972615815433211238, 5.15165993315668965696796025589, 5.53104872938569287196583548776, 6.08476828479604175178802693467, 6.18336546061230623342307495037, 6.58841792152257971985007080180, 7.38054044225898634615941577805, 7.73441976365842372781486087829, 8.465615571691745927792215072143, 8.775126213065446762474526089106, 9.062811038441739238282432200803, 9.532557728911292126627635924558, 9.976447831725577390742809063318, 10.22320619755545118655195289616
