| L(s) = 1 | + 4-s + 4·7-s + 12·11-s − 3·16-s − 6·17-s + 8·19-s + 12·23-s − 2·25-s + 4·28-s − 12·29-s + 4·31-s + 12·44-s − 2·49-s − 18·53-s − 7·64-s − 28·67-s − 6·68-s + 8·76-s + 48·77-s + 12·83-s − 30·89-s + 12·92-s + 22·97-s − 2·100-s + 20·103-s + 10·109-s − 12·112-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.51·7-s + 3.61·11-s − 3/4·16-s − 1.45·17-s + 1.83·19-s + 2.50·23-s − 2/5·25-s + 0.755·28-s − 2.22·29-s + 0.718·31-s + 1.80·44-s − 2/7·49-s − 2.47·53-s − 7/8·64-s − 3.42·67-s − 0.727·68-s + 0.917·76-s + 5.47·77-s + 1.31·83-s − 3.17·89-s + 1.25·92-s + 2.23·97-s − 1/5·100-s + 1.97·103-s + 0.957·109-s − 1.13·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.731640011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.731640011\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 31 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 139 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
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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
−10.73552018675074430738703220946, −9.645096010915611562991231968182, −9.476605558325649063804558491292, −9.201509492856631165075593383078, −8.767674385179784012514615145752, −8.581271984992672113834928231185, −7.64138108027541638058023826948, −7.35974939544665156116596977847, −7.05491958347309582255858622265, −6.50035462255486572978613942605, −6.27137472878734127817790021153, −5.64765497079532599249654896213, −4.90382737578548711706686072427, −4.44845375668621941487040205351, −4.37290338351873368521713976011, −3.27920306185655880357948539361, −3.27424464561895398607380194715, −1.91912924591334761207820451529, −1.61666644346413186719296289506, −1.12103771812187319926668138464,
1.12103771812187319926668138464, 1.61666644346413186719296289506, 1.91912924591334761207820451529, 3.27424464561895398607380194715, 3.27920306185655880357948539361, 4.37290338351873368521713976011, 4.44845375668621941487040205351, 4.90382737578548711706686072427, 5.64765497079532599249654896213, 6.27137472878734127817790021153, 6.50035462255486572978613942605, 7.05491958347309582255858622265, 7.35974939544665156116596977847, 7.64138108027541638058023826948, 8.581271984992672113834928231185, 8.767674385179784012514615145752, 9.201509492856631165075593383078, 9.476605558325649063804558491292, 9.645096010915611562991231968182, 10.73552018675074430738703220946
