| L(s) = 1 | − 4-s − 5-s − 7-s − 9-s + 3·11-s − 5·13-s + 16-s + 17-s − 3·19-s + 20-s + 8·23-s + 25-s + 28-s − 7·29-s − 2·31-s + 35-s + 36-s + 3·37-s − 3·41-s + 8·43-s − 3·44-s + 45-s + 6·47-s − 3·49-s + 5·52-s − 3·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.447·5-s − 0.377·7-s − 1/3·9-s + 0.904·11-s − 1.38·13-s + 1/4·16-s + 0.242·17-s − 0.688·19-s + 0.223·20-s + 1.66·23-s + 1/5·25-s + 0.188·28-s − 1.29·29-s − 0.359·31-s + 0.169·35-s + 1/6·36-s + 0.493·37-s − 0.468·41-s + 1.21·43-s − 0.452·44-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.693·52-s − 0.412·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 57 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 86 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 76 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 104 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 19 T + 228 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 61 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 52 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−13.4105417272, −13.0008260426, −12.7262233156, −12.3764639817, −11.8146299259, −11.4945672373, −11.0388138923, −10.5855466782, −10.0773652034, −9.53497990954, −9.22027950082, −8.91192830585, −8.40697537192, −7.72967409056, −7.39423280235, −6.92641634851, −6.45784585764, −5.77994190657, −5.25368801550, −4.82386896711, −4.06829864466, −3.79982467476, −2.94233008584, −2.40874934349, −1.24145428098, 0,
1.24145428098, 2.40874934349, 2.94233008584, 3.79982467476, 4.06829864466, 4.82386896711, 5.25368801550, 5.77994190657, 6.45784585764, 6.92641634851, 7.39423280235, 7.72967409056, 8.40697537192, 8.91192830585, 9.22027950082, 9.53497990954, 10.0773652034, 10.5855466782, 11.0388138923, 11.4945672373, 11.8146299259, 12.3764639817, 12.7262233156, 13.0008260426, 13.4105417272
