| L(s) = 1 | − 4·4-s + 9-s − 2·13-s + 12·16-s − 17-s − 2·19-s − 25-s − 4·36-s − 14·43-s − 12·47-s + 2·49-s + 8·52-s − 12·53-s + 12·59-s − 32·64-s − 8·67-s + 4·68-s + 8·76-s + 81-s − 12·83-s + 4·100-s + 10·103-s − 2·117-s − 13·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1/3·9-s − 0.554·13-s + 3·16-s − 0.242·17-s − 0.458·19-s − 1/5·25-s − 2/3·36-s − 2.13·43-s − 1.75·47-s + 2/7·49-s + 1.10·52-s − 1.64·53-s + 1.56·59-s − 4·64-s − 0.977·67-s + 0.485·68-s + 0.917·76-s + 1/9·81-s − 1.31·83-s + 2/5·100-s + 0.985·103-s − 0.184·117-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44217 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44217 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−9.721668236319765682540445072382, −9.642727340101070443548067847258, −8.747358907070908179484393186464, −8.618966621189813626434174913816, −7.960547248828429430124687340941, −7.53511853023675535658862215085, −6.66463575826783700250676263362, −6.12714514378066180257024905951, −5.14306131647545403565477997005, −5.04692212052117448633199558052, −4.31112644939450006748261353184, −3.76588854630918978975078187598, −3.00632648437536992753290178800, −1.57155610960796422893905590944, 0,
1.57155610960796422893905590944, 3.00632648437536992753290178800, 3.76588854630918978975078187598, 4.31112644939450006748261353184, 5.04692212052117448633199558052, 5.14306131647545403565477997005, 6.12714514378066180257024905951, 6.66463575826783700250676263362, 7.53511853023675535658862215085, 7.960547248828429430124687340941, 8.618966621189813626434174913816, 8.747358907070908179484393186464, 9.642727340101070443548067847258, 9.721668236319765682540445072382
