| L(s) = 1 | − 3-s − 3·4-s + 9-s − 8·11-s + 3·12-s + 5·16-s + 4·17-s + 8·19-s + 25-s − 27-s − 4·29-s + 8·33-s − 3·36-s + 24·44-s − 5·48-s − 14·49-s − 4·51-s − 20·53-s − 8·57-s − 3·64-s + 24·67-s − 12·68-s − 75-s − 24·76-s + 81-s + 24·83-s + 4·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s + 1/3·9-s − 2.41·11-s + 0.866·12-s + 5/4·16-s + 0.970·17-s + 1.83·19-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 1.39·33-s − 1/2·36-s + 3.61·44-s − 0.721·48-s − 2·49-s − 0.560·51-s − 2.74·53-s − 1.05·57-s − 3/8·64-s + 2.93·67-s − 1.45·68-s − 0.115·75-s − 2.75·76-s + 1/9·81-s + 2.63·83-s + 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 31 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.114900408024103140014243113668, −7.66488013441745380243230842523, −7.62168634080583778803253039703, −6.76366529686096520107036215669, −6.22644927255442252578289500040, −5.48779277170674803346725706409, −5.23920392624592057055772361749, −5.11085823439540905400666440111, −4.62012066537284998425540488776, −3.83645868031632569816966802271, −3.25332827792826642629963089952, −2.91451080617330683387950783522, −1.86030328865315941936552250136, −0.850059814000557593825806016483, 0,
0.850059814000557593825806016483, 1.86030328865315941936552250136, 2.91451080617330683387950783522, 3.25332827792826642629963089952, 3.83645868031632569816966802271, 4.62012066537284998425540488776, 5.11085823439540905400666440111, 5.23920392624592057055772361749, 5.48779277170674803346725706409, 6.22644927255442252578289500040, 6.76366529686096520107036215669, 7.62168634080583778803253039703, 7.66488013441745380243230842523, 8.114900408024103140014243113668
