| L(s) = 1 | + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 5·9-s − 12·11-s − 8·14-s + 5·16-s − 10·18-s − 24·22-s − 6·23-s − 25-s − 12·28-s + 6·32-s − 15·36-s − 20·37-s − 2·43-s − 36·44-s − 12·46-s + 9·49-s − 2·50-s + 18·53-s − 16·56-s + 20·63-s + 7·64-s − 8·67-s − 12·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 5/3·9-s − 3.61·11-s − 2.13·14-s + 5/4·16-s − 2.35·18-s − 5.11·22-s − 1.25·23-s − 1/5·25-s − 2.26·28-s + 1.06·32-s − 5/2·36-s − 3.28·37-s − 0.304·43-s − 5.42·44-s − 1.76·46-s + 9/7·49-s − 0.282·50-s + 2.47·53-s − 2.13·56-s + 2.51·63-s + 7/8·64-s − 0.977·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1552516 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1552516 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−7.36410689745433254804873728837, −7.05985684621185329175912053099, −6.33514500797210537341254603766, −5.94744023022306454130283790935, −5.49227575392363826183663580413, −5.48197881161062206485371067956, −4.99949740842160884734639303520, −4.36896427817318943075761707833, −3.55398942164639159030341252985, −3.31534392384618004485979133214, −2.78977260166977870939080820706, −2.51994269176562419526281033451, −2.01410830691968436892778575899, 0, 0,
2.01410830691968436892778575899, 2.51994269176562419526281033451, 2.78977260166977870939080820706, 3.31534392384618004485979133214, 3.55398942164639159030341252985, 4.36896427817318943075761707833, 4.99949740842160884734639303520, 5.48197881161062206485371067956, 5.49227575392363826183663580413, 5.94744023022306454130283790935, 6.33514500797210537341254603766, 7.05985684621185329175912053099, 7.36410689745433254804873728837
