| L(s) = 1 | − 2·2-s − 3·3-s + 3·4-s − 4·5-s + 6·6-s − 2·7-s − 4·8-s + 4·9-s + 8·10-s − 6·11-s − 9·12-s + 4·14-s + 12·15-s + 5·16-s − 4·17-s − 8·18-s − 2·19-s − 12·20-s + 6·21-s + 12·22-s − 4·23-s + 12·24-s + 2·25-s − 6·28-s − 8·29-s − 24·30-s − 4·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s − 1.78·5-s + 2.44·6-s − 0.755·7-s − 1.41·8-s + 4/3·9-s + 2.52·10-s − 1.80·11-s − 2.59·12-s + 1.06·14-s + 3.09·15-s + 5/4·16-s − 0.970·17-s − 1.88·18-s − 0.458·19-s − 2.68·20-s + 1.30·21-s + 2.55·22-s − 0.834·23-s + 2.44·24-s + 2/5·25-s − 1.13·28-s − 1.48·29-s − 4.38·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20172 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20172 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−16.1440817264, −16.0980463378, −15.5438761265, −15.2322872584, −14.8034113429, −13.5603503236, −13.1338784795, −12.3632258252, −12.3535797825, −11.5374061458, −11.2937627155, −10.8687735754, −10.5945207017, −9.81085245169, −9.48966114694, −8.46774633173, −8.02435745103, −7.75573819994, −6.85698182335, −6.70748925180, −5.69856539644, −5.36274716754, −4.29114537424, −3.54075497824, −2.32002779008, 0, 0,
2.32002779008, 3.54075497824, 4.29114537424, 5.36274716754, 5.69856539644, 6.70748925180, 6.85698182335, 7.75573819994, 8.02435745103, 8.46774633173, 9.48966114694, 9.81085245169, 10.5945207017, 10.8687735754, 11.2937627155, 11.5374061458, 12.3535797825, 12.3632258252, 13.1338784795, 13.5603503236, 14.8034113429, 15.2322872584, 15.5438761265, 16.0980463378, 16.1440817264
