| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8·7-s − 8-s + 9-s + 10-s + 12-s + 8·14-s − 15-s + 16-s + 12·17-s − 18-s − 20-s − 8·21-s − 24-s + 25-s + 27-s − 8·28-s + 30-s − 32-s − 12·34-s + 8·35-s + 36-s + 40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 3.02·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 2.13·14-s − 0.258·15-s + 1/4·16-s + 2.91·17-s − 0.235·18-s − 0.223·20-s − 1.74·21-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 1.51·28-s + 0.182·30-s − 0.176·32-s − 2.05·34-s + 1.35·35-s + 1/6·36-s + 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.305587122869086271308905422277, −9.116818239719236618689703406908, −8.101485244373939343160290559425, −7.941231322257043910223245152113, −7.35042654382863533673550380313, −6.78354731906038205235867209347, −6.42217617652666865799421983967, −5.87146842228725166509134997887, −5.34762325653913158633333815914, −4.22720411424127134624171462120, −3.36585804145949552210873743484, −3.25067910161980170709704604527, −2.84728890251355077691674063068, −1.34557356174820418480074519421, 0,
1.34557356174820418480074519421, 2.84728890251355077691674063068, 3.25067910161980170709704604527, 3.36585804145949552210873743484, 4.22720411424127134624171462120, 5.34762325653913158633333815914, 5.87146842228725166509134997887, 6.42217617652666865799421983967, 6.78354731906038205235867209347, 7.35042654382863533673550380313, 7.941231322257043910223245152113, 8.101485244373939343160290559425, 9.116818239719236618689703406908, 9.305587122869086271308905422277
