| L(s) = 1 | + 2-s − 4-s − 3·8-s − 2·9-s + 11-s + 13-s − 16-s − 3·17-s − 2·18-s + 22-s − 2·25-s + 26-s − 3·27-s + 7·29-s − 6·31-s + 5·32-s − 3·34-s + 2·36-s − 9·41-s + 2·43-s − 44-s − 9·47-s + 49-s − 2·50-s − 52-s − 3·54-s + 7·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 2/3·9-s + 0.301·11-s + 0.277·13-s − 1/4·16-s − 0.727·17-s − 0.471·18-s + 0.213·22-s − 2/5·25-s + 0.196·26-s − 0.577·27-s + 1.29·29-s − 1.07·31-s + 0.883·32-s − 0.514·34-s + 1/3·36-s − 1.40·41-s + 0.304·43-s − 0.150·44-s − 1.31·47-s + 1/7·49-s − 0.282·50-s − 0.138·52-s − 0.408·54-s + 0.919·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 509232 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 509232 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.515814364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.515814364\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 103 | $C_2$ | \( 1 - 12 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.455351188243329479711167507235, −8.203283203199093948328203330554, −7.70437040418849221454070184818, −6.90494501214452960813805121103, −6.57657600833347725302925121119, −6.20905110842876822335460540754, −5.57253518513294466155545440219, −5.20149568407039243876175049044, −4.78906778061469194558691607940, −4.14822893124892873374419319467, −3.63887115357959521901772105461, −3.28198282062809047428450680528, −2.49827631201068496843971761004, −1.82746422605243952409172203860, −0.57180449554849453248108816773,
0.57180449554849453248108816773, 1.82746422605243952409172203860, 2.49827631201068496843971761004, 3.28198282062809047428450680528, 3.63887115357959521901772105461, 4.14822893124892873374419319467, 4.78906778061469194558691607940, 5.20149568407039243876175049044, 5.57253518513294466155545440219, 6.20905110842876822335460540754, 6.57657600833347725302925121119, 6.90494501214452960813805121103, 7.70437040418849221454070184818, 8.203283203199093948328203330554, 8.455351188243329479711167507235
