| L(s) = 1 | + 4·3-s − 3·4-s − 4·5-s + 6·9-s + 11-s − 12·12-s − 16·15-s + 5·16-s + 12·20-s − 8·23-s + 2·25-s − 4·27-s + 20·31-s + 4·33-s − 18·36-s − 12·37-s − 3·44-s − 24·45-s − 20·47-s + 20·48-s + 49-s − 12·53-s − 4·55-s + 4·59-s + 48·60-s − 3·64-s + 16·67-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 3/2·4-s − 1.78·5-s + 2·9-s + 0.301·11-s − 3.46·12-s − 4.13·15-s + 5/4·16-s + 2.68·20-s − 1.66·23-s + 2/5·25-s − 0.769·27-s + 3.59·31-s + 0.696·33-s − 3·36-s − 1.97·37-s − 0.452·44-s − 3.57·45-s − 2.91·47-s + 2.88·48-s + 1/7·49-s − 1.64·53-s − 0.539·55-s + 0.520·59-s + 6.19·60-s − 3/8·64-s + 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65219 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.742618950410792459548371471551, −8.678915900547504955372461266927, −8.572504027942197515704035369998, −8.339912824949481407935374648788, −7.82867506007281568614416512666, −7.67378178112872223010860918103, −6.70583207072298392541889211635, −5.96443978110022033983848919969, −4.94559945005359926667122742610, −4.28286203006294711588571475853, −4.07930883034575373640745113861, −3.26888344602981049454952325554, −3.17103530919462504679231815837, −1.91939450772296709675161614006, 0,
1.91939450772296709675161614006, 3.17103530919462504679231815837, 3.26888344602981049454952325554, 4.07930883034575373640745113861, 4.28286203006294711588571475853, 4.94559945005359926667122742610, 5.96443978110022033983848919969, 6.70583207072298392541889211635, 7.67378178112872223010860918103, 7.82867506007281568614416512666, 8.339912824949481407935374648788, 8.572504027942197515704035369998, 8.678915900547504955372461266927, 9.742618950410792459548371471551
