| L(s) = 1 | − 3-s − 4·4-s − 4·7-s + 6·11-s + 4·12-s + 2·13-s + 12·16-s − 12·17-s + 19-s + 4·21-s − 12·23-s + 5·25-s + 27-s + 16·28-s − 6·29-s − 5·31-s − 6·33-s − 2·37-s − 2·39-s + 4·43-s − 24·44-s − 6·47-s − 12·48-s + 9·49-s + 12·51-s − 8·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2·4-s − 1.51·7-s + 1.80·11-s + 1.15·12-s + 0.554·13-s + 3·16-s − 2.91·17-s + 0.229·19-s + 0.872·21-s − 2.50·23-s + 25-s + 0.192·27-s + 3.02·28-s − 1.11·29-s − 0.898·31-s − 1.04·33-s − 0.328·37-s − 0.320·39-s + 0.609·43-s − 3.61·44-s − 0.875·47-s − 1.73·48-s + 9/7·49-s + 1.68·51-s − 1.10·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3251731158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3251731158\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−12.43910921535886099413207746231, −11.44611654484394909958756734412, −11.43451544201108740016613951368, −10.67916768698718852694132936098, −10.03338017554999544962100238572, −9.668201765387804100803087951291, −9.238726619268364511640779083065, −8.954477574557389266822481880619, −8.574753939850581975155307663895, −7.978813380158801354819627663218, −7.02912017143054913870090716736, −6.39530755092780552400217075952, −6.34412844073955488095690382710, −5.59153526927642284869914102854, −4.92613828564084320850381226945, −4.11700446424569758226342094974, −3.95070459286534361950933181856, −3.46599951897147741034859237712, −1.98576064178313417115910997199, −0.46472755962556151577273918472,
0.46472755962556151577273918472, 1.98576064178313417115910997199, 3.46599951897147741034859237712, 3.95070459286534361950933181856, 4.11700446424569758226342094974, 4.92613828564084320850381226945, 5.59153526927642284869914102854, 6.34412844073955488095690382710, 6.39530755092780552400217075952, 7.02912017143054913870090716736, 7.978813380158801354819627663218, 8.574753939850581975155307663895, 8.954477574557389266822481880619, 9.238726619268364511640779083065, 9.668201765387804100803087951291, 10.03338017554999544962100238572, 10.67916768698718852694132936098, 11.43451544201108740016613951368, 11.44611654484394909958756734412, 12.43910921535886099413207746231
