| L(s) = 1 | + 2·2-s − 3-s − 4-s − 2·6-s − 8·8-s + 9-s + 11-s + 12-s − 7·16-s − 4·17-s + 2·18-s + 2·22-s + 8·24-s − 6·25-s − 27-s − 12·29-s − 16·31-s + 14·32-s − 33-s − 8·34-s − 36-s + 12·37-s − 4·41-s − 44-s + 7·48-s + 2·49-s − 12·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s − 1/2·4-s − 0.816·6-s − 2.82·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 7/4·16-s − 0.970·17-s + 0.471·18-s + 0.426·22-s + 1.63·24-s − 6/5·25-s − 0.192·27-s − 2.22·29-s − 2.87·31-s + 2.47·32-s − 0.174·33-s − 1.37·34-s − 1/6·36-s + 1.97·37-s − 0.624·41-s − 0.150·44-s + 1.01·48-s + 2/7·49-s − 1.69·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35937 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35937 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−9.979784146917935153776686794495, −9.531112831864384789210057232688, −8.975261543279013968212899152873, −8.938244517857001424659052468572, −7.77744707656776841753907290662, −7.46995816184461358823952634116, −6.38866021531112361683475479825, −6.11982556539470572208453322290, −5.31622482563829752699303340190, −5.25909302550772251578024339366, −4.20407903283329927801590362766, −3.98994534926773298366607994601, −3.36662241945177810035682717763, −2.09666333780189433867543562159, 0,
2.09666333780189433867543562159, 3.36662241945177810035682717763, 3.98994534926773298366607994601, 4.20407903283329927801590362766, 5.25909302550772251578024339366, 5.31622482563829752699303340190, 6.11982556539470572208453322290, 6.38866021531112361683475479825, 7.46995816184461358823952634116, 7.77744707656776841753907290662, 8.938244517857001424659052468572, 8.975261543279013968212899152873, 9.531112831864384789210057232688, 9.979784146917935153776686794495
