| L(s) = 1 | − 2-s − 4-s + 5-s − 3·7-s + 3·8-s − 10-s + 3·11-s + 4·13-s + 3·14-s − 16-s − 8·17-s − 2·19-s − 20-s − 3·22-s − 5·25-s − 4·26-s + 3·28-s + 7·29-s − 4·31-s − 5·32-s + 8·34-s − 3·35-s + 2·38-s + 3·40-s + 3·41-s − 43-s − 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.13·7-s + 1.06·8-s − 0.316·10-s + 0.904·11-s + 1.10·13-s + 0.801·14-s − 1/4·16-s − 1.94·17-s − 0.458·19-s − 0.223·20-s − 0.639·22-s − 25-s − 0.784·26-s + 0.566·28-s + 1.29·29-s − 0.718·31-s − 0.883·32-s + 1.37·34-s − 0.507·35-s + 0.324·38-s + 0.474·40-s + 0.468·41-s − 0.152·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128812 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128812 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 32203 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 215 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 49 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 72 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7 T + 88 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 26 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T - 41 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 76 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−13.8102140746, −13.6160582439, −13.1606051300, −12.8895600066, −12.5088949548, −11.6948714679, −11.3602675305, −10.8524753569, −10.5145822291, −9.83239783173, −9.56137589160, −9.25526281107, −8.72093078746, −8.33363480831, −7.96440927361, −6.99495289090, −6.58917039176, −6.41574278362, −5.74193046401, −4.98057196589, −4.28384446290, −3.91036546302, −3.20779091750, −2.18959784870, −1.38932369524, 0,
1.38932369524, 2.18959784870, 3.20779091750, 3.91036546302, 4.28384446290, 4.98057196589, 5.74193046401, 6.41574278362, 6.58917039176, 6.99495289090, 7.96440927361, 8.33363480831, 8.72093078746, 9.25526281107, 9.56137589160, 9.83239783173, 10.5145822291, 10.8524753569, 11.3602675305, 11.6948714679, 12.5088949548, 12.8895600066, 13.1606051300, 13.6160582439, 13.8102140746
