| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 5-s − 4·6-s − 4·8-s + 3·9-s + 2·10-s − 8·11-s + 6·12-s + 13-s − 2·15-s + 5·16-s + 3·17-s − 6·18-s − 3·20-s + 16·22-s − 8·23-s − 8·24-s − 8·25-s − 2·26-s + 4·27-s − 11·29-s + 4·30-s + 12·31-s − 6·32-s − 16·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.447·5-s − 1.63·6-s − 1.41·8-s + 9-s + 0.632·10-s − 2.41·11-s + 1.73·12-s + 0.277·13-s − 0.516·15-s + 5/4·16-s + 0.727·17-s − 1.41·18-s − 0.670·20-s + 3.41·22-s − 1.66·23-s − 1.63·24-s − 8/5·25-s − 0.392·26-s + 0.769·27-s − 2.04·29-s + 0.730·30-s + 2.15·31-s − 1.06·32-s − 2.78·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4691556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4691556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - T + 25 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 11 T + 77 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 69 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 77 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 161 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 17 T + 193 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3 T - 63 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 23 T + 309 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T + 49 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−8.639104015306652654046022550134, −8.319919433686698421838341544832, −8.024977885173213731235973985050, −7.963052943678494101214506545683, −7.58423129413612126555166449990, −7.46426151540045305385790182310, −6.53844986343870282741077518178, −6.39515225940001507836900406469, −5.70406986855634074566283277660, −5.55221809243616450887418830939, −4.58739518288769493766585214665, −4.58247138454900292960903380328, −3.53448299680613733657951785136, −3.43602963638392449961245525625, −2.78222236276994469773121213760, −2.48337630399636066051273007451, −1.74217653237120005055592286485, −1.52329535296591515958877936932, 0, 0,
1.52329535296591515958877936932, 1.74217653237120005055592286485, 2.48337630399636066051273007451, 2.78222236276994469773121213760, 3.43602963638392449961245525625, 3.53448299680613733657951785136, 4.58247138454900292960903380328, 4.58739518288769493766585214665, 5.55221809243616450887418830939, 5.70406986855634074566283277660, 6.39515225940001507836900406469, 6.53844986343870282741077518178, 7.46426151540045305385790182310, 7.58423129413612126555166449990, 7.963052943678494101214506545683, 8.024977885173213731235973985050, 8.319919433686698421838341544832, 8.639104015306652654046022550134
