| L(s) = 1 | − 2·4-s + 6·5-s + 9-s − 2·13-s + 4·16-s − 2·17-s − 12·20-s + 17·25-s + 12·29-s − 2·36-s − 8·37-s − 6·41-s + 6·45-s + 2·49-s + 4·52-s − 12·53-s + 16·61-s − 8·64-s − 12·65-s + 4·68-s + 4·73-s + 24·80-s + 81-s − 12·85-s − 32·97-s − 34·100-s + 40·109-s + ⋯ |
| L(s) = 1 | − 4-s + 2.68·5-s + 1/3·9-s − 0.554·13-s + 16-s − 0.485·17-s − 2.68·20-s + 17/5·25-s + 2.22·29-s − 1/3·36-s − 1.31·37-s − 0.937·41-s + 0.894·45-s + 2/7·49-s + 0.554·52-s − 1.64·53-s + 2.04·61-s − 64-s − 1.48·65-s + 0.485·68-s + 0.468·73-s + 2.68·80-s + 1/9·81-s − 1.30·85-s − 3.24·97-s − 3.39·100-s + 3.83·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.679417732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.679417732\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−10.23827402727964604848387184191, −9.642727340101070443548067847258, −9.519241921389692003197336146870, −8.618966621189813626434174913816, −8.610578703883184504620815355064, −7.66773143025981285351749668682, −6.66463575826783700250676263362, −6.56329579539546958275321552423, −5.81241206259765754250513501442, −5.14306131647545403565477997005, −5.01602847451418763245834802500, −4.10168552133706270535583651662, −3.00632648437536992753290178800, −2.22032145426687050578358306238, −1.39440035412653237109928920217,
1.39440035412653237109928920217, 2.22032145426687050578358306238, 3.00632648437536992753290178800, 4.10168552133706270535583651662, 5.01602847451418763245834802500, 5.14306131647545403565477997005, 5.81241206259765754250513501442, 6.56329579539546958275321552423, 6.66463575826783700250676263362, 7.66773143025981285351749668682, 8.610578703883184504620815355064, 8.618966621189813626434174913816, 9.519241921389692003197336146870, 9.642727340101070443548067847258, 10.23827402727964604848387184191
