| L(s) = 1 | + 4·5-s − 8·11-s + 8·19-s + 11·25-s + 8·29-s + 16·41-s − 2·49-s − 32·55-s − 24·59-s + 4·61-s + 16·71-s + 16·79-s + 32·95-s − 24·101-s − 36·109-s + 26·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.41·11-s + 1.83·19-s + 11/5·25-s + 1.48·29-s + 2.49·41-s − 2/7·49-s − 4.31·55-s − 3.12·59-s + 0.512·61-s + 1.89·71-s + 1.80·79-s + 3.28·95-s − 2.38·101-s − 3.44·109-s + 2.36·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.069730878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.069730878\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.65879579346267356258879884230, −10.83570633696793494852455427576, −10.70732716903024852763786465084, −10.43355746923322554872980265295, −9.711675974419078154307610084218, −9.435168377190757686636595233818, −9.245279456768092535170402034086, −8.299380074429026524925732610899, −7.82619478472493033762344294927, −7.66822164365308683845383546332, −6.75750391678551577095774735061, −6.39229219938449426721449446526, −5.63478898908345632250485287133, −5.38636321607616423847974741462, −5.03965615536919006739960089996, −4.32417740296710715670041851557, −2.98152439754895514848236179863, −2.85451344586925836438889587997, −2.14246318192079397400286099028, −1.05956648832118516673276067628,
1.05956648832118516673276067628, 2.14246318192079397400286099028, 2.85451344586925836438889587997, 2.98152439754895514848236179863, 4.32417740296710715670041851557, 5.03965615536919006739960089996, 5.38636321607616423847974741462, 5.63478898908345632250485287133, 6.39229219938449426721449446526, 6.75750391678551577095774735061, 7.66822164365308683845383546332, 7.82619478472493033762344294927, 8.299380074429026524925732610899, 9.245279456768092535170402034086, 9.435168377190757686636595233818, 9.711675974419078154307610084218, 10.43355746923322554872980265295, 10.70732716903024852763786465084, 10.83570633696793494852455427576, 11.65879579346267356258879884230
