| L(s) = 1 | − 4·5-s − 5·7-s − 6·13-s + 4·17-s + 7·19-s + 4·23-s + 5·25-s − 16·29-s − 5·31-s + 20·35-s − 3·37-s + 16·41-s − 22·43-s + 4·47-s + 18·49-s − 4·53-s + 12·59-s + 2·61-s + 24·65-s − 3·67-s − 24·71-s − 73-s + 79-s − 24·83-s − 16·85-s − 8·89-s + 30·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.88·7-s − 1.66·13-s + 0.970·17-s + 1.60·19-s + 0.834·23-s + 25-s − 2.97·29-s − 0.898·31-s + 3.38·35-s − 0.493·37-s + 2.49·41-s − 3.35·43-s + 0.583·47-s + 18/7·49-s − 0.549·53-s + 1.56·59-s + 0.256·61-s + 2.97·65-s − 0.366·67-s − 2.84·71-s − 0.117·73-s + 0.112·79-s − 2.63·83-s − 1.73·85-s − 0.847·89-s + 3.14·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 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
−9.816837575317312139031826969606, −9.488019967332540966935454192217, −8.916831033182143863108953580597, −8.672572862635385294877753235305, −7.73249703539176369559659548647, −7.59601618758351430060709613098, −7.28463351284945269848124155424, −7.13175620284314832729278167324, −6.50082714280516506194874059167, −5.71629900459311173900762353035, −5.45767443891443563886328149961, −5.09011708905491618833734717326, −4.07473204626147215665434951344, −4.01214972767985722727138792610, −3.24326086864043172269554851350, −3.19798615536227075639595940484, −2.49061059316318362188275236199, −1.35666611484060859701475535105, 0, 0,
1.35666611484060859701475535105, 2.49061059316318362188275236199, 3.19798615536227075639595940484, 3.24326086864043172269554851350, 4.01214972767985722727138792610, 4.07473204626147215665434951344, 5.09011708905491618833734717326, 5.45767443891443563886328149961, 5.71629900459311173900762353035, 6.50082714280516506194874059167, 7.13175620284314832729278167324, 7.28463351284945269848124155424, 7.59601618758351430060709613098, 7.73249703539176369559659548647, 8.672572862635385294877753235305, 8.916831033182143863108953580597, 9.488019967332540966935454192217, 9.816837575317312139031826969606
