| L(s) = 1 | − 3·3-s + 4-s − 4·7-s + 6·9-s − 3·12-s + 6·13-s + 16-s − 16·19-s + 12·21-s − 25-s − 9·27-s − 4·28-s + 6·31-s + 6·36-s − 16·37-s − 18·39-s + 14·43-s − 3·48-s − 2·49-s + 6·52-s + 48·57-s + 8·61-s − 24·63-s + 64-s − 8·67-s − 24·73-s + 3·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1/2·4-s − 1.51·7-s + 2·9-s − 0.866·12-s + 1.66·13-s + 1/4·16-s − 3.67·19-s + 2.61·21-s − 1/5·25-s − 1.73·27-s − 0.755·28-s + 1.07·31-s + 36-s − 2.63·37-s − 2.88·39-s + 2.13·43-s − 0.433·48-s − 2/7·49-s + 0.832·52-s + 6.35·57-s + 1.02·61-s − 3.02·63-s + 1/8·64-s − 0.977·67-s − 2.80·73-s + 0.346·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30276 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30276 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−10.46054334513251190747035603866, −10.12617761918087977800848259614, −9.236533170962776009348455496651, −8.643506374817898327978246248838, −8.246512694538120784323368196643, −7.09229495493052892657316288498, −6.73575070734888517320889411881, −6.21803091761568248529227009695, −6.11755064298029935226028768008, −5.43283521764238035821195525394, −4.22146327216160305586457647308, −4.11203255589147381479537694825, −2.94700675923958365463547161100, −1.68307127909684975914702090920, 0,
1.68307127909684975914702090920, 2.94700675923958365463547161100, 4.11203255589147381479537694825, 4.22146327216160305586457647308, 5.43283521764238035821195525394, 6.11755064298029935226028768008, 6.21803091761568248529227009695, 6.73575070734888517320889411881, 7.09229495493052892657316288498, 8.246512694538120784323368196643, 8.643506374817898327978246248838, 9.236533170962776009348455496651, 10.12617761918087977800848259614, 10.46054334513251190747035603866
