| L(s) = 1 | + 3-s + 2·4-s − 4·5-s + 5·7-s − 4·11-s + 2·12-s + 2·13-s − 4·15-s − 17-s − 19-s − 8·20-s + 5·21-s − 2·23-s + 5·25-s − 27-s + 10·28-s + 12·29-s + 9·31-s − 4·33-s − 20·35-s + 11·37-s + 2·39-s + 20·41-s − 14·43-s − 8·44-s − 6·47-s + 18·49-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 4-s − 1.78·5-s + 1.88·7-s − 1.20·11-s + 0.577·12-s + 0.554·13-s − 1.03·15-s − 0.242·17-s − 0.229·19-s − 1.78·20-s + 1.09·21-s − 0.417·23-s + 25-s − 0.192·27-s + 1.88·28-s + 2.22·29-s + 1.61·31-s − 0.696·33-s − 3.38·35-s + 1.80·37-s + 0.320·39-s + 3.12·41-s − 2.13·43-s − 1.20·44-s − 0.875·47-s + 18/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127449 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.088574840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.088574840\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−11.50527704670798145229390408772, −11.30531768196588505589621308739, −11.06582916896347477359951887421, −10.31105392358276523595680515967, −10.23229844526700693614003125108, −9.175914912890246868750902596872, −8.596280966887225024299770787904, −8.196019431663581185684440422620, −7.925707102100040371154447497675, −7.68870739423632951192390992738, −7.28061346355928208588970579317, −6.31160996963708251926822914662, −6.15159839579106948738444959959, −5.03748234314788334967738981615, −4.46291870338408115094853555768, −4.41591920375236971210802932676, −3.41441088176636134159571646995, −2.64828239875032214219152203457, −2.26026979282041020628987529760, −1.02909279819066640086988665290,
1.02909279819066640086988665290, 2.26026979282041020628987529760, 2.64828239875032214219152203457, 3.41441088176636134159571646995, 4.41591920375236971210802932676, 4.46291870338408115094853555768, 5.03748234314788334967738981615, 6.15159839579106948738444959959, 6.31160996963708251926822914662, 7.28061346355928208588970579317, 7.68870739423632951192390992738, 7.925707102100040371154447497675, 8.196019431663581185684440422620, 8.596280966887225024299770787904, 9.175914912890246868750902596872, 10.23229844526700693614003125108, 10.31105392358276523595680515967, 11.06582916896347477359951887421, 11.30531768196588505589621308739, 11.50527704670798145229390408772
