| L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s + 2·11-s − 2·13-s − 4·15-s − 6·17-s + 4·19-s − 8·23-s − 2·25-s + 4·27-s − 2·29-s − 2·31-s + 4·33-s − 8·37-s − 4·39-s + 16·41-s + 2·43-s − 6·45-s − 8·47-s − 14·49-s − 12·51-s − 4·53-s − 4·55-s + 8·57-s − 12·59-s + 4·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s + 0.603·11-s − 0.554·13-s − 1.03·15-s − 1.45·17-s + 0.917·19-s − 1.66·23-s − 2/5·25-s + 0.769·27-s − 0.371·29-s − 0.359·31-s + 0.696·33-s − 1.31·37-s − 0.640·39-s + 2.49·41-s + 0.304·43-s − 0.894·45-s − 1.16·47-s − 2·49-s − 1.68·51-s − 0.549·53-s − 0.539·55-s + 1.05·57-s − 1.56·59-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 162 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
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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
−7.77432528993576502899890469810, −7.65425829623518263113816326015, −7.20506580304903105327459960488, −6.82221858406693904008261199301, −6.37518934610204753834785880523, −6.31614143362224937156444773543, −5.57812386120543754419872677516, −5.34544709434699038567356596908, −4.60491340601679432565901574917, −4.55305562941864657064600616535, −4.08509574251541827073450200689, −3.77914871621928273782736500567, −3.40661110317308954648117257619, −3.05653036017541197730231224156, −2.41368127368505185434494847011, −2.21489667621299387035845749229, −1.59744671188861808410776280296, −1.24528761339727409514722328873, 0, 0,
1.24528761339727409514722328873, 1.59744671188861808410776280296, 2.21489667621299387035845749229, 2.41368127368505185434494847011, 3.05653036017541197730231224156, 3.40661110317308954648117257619, 3.77914871621928273782736500567, 4.08509574251541827073450200689, 4.55305562941864657064600616535, 4.60491340601679432565901574917, 5.34544709434699038567356596908, 5.57812386120543754419872677516, 6.31614143362224937156444773543, 6.37518934610204753834785880523, 6.82221858406693904008261199301, 7.20506580304903105327459960488, 7.65425829623518263113816326015, 7.77432528993576502899890469810
