| L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 2·6-s − 8·7-s − 4·8-s + 9-s − 3·12-s + 16·14-s + 5·16-s − 2·18-s + 19-s + 8·21-s + 4·24-s − 10·25-s − 27-s − 24·28-s − 12·29-s − 6·32-s + 3·36-s − 2·38-s − 12·41-s − 16·42-s − 8·43-s − 5·48-s + 34·49-s + 20·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.816·6-s − 3.02·7-s − 1.41·8-s + 1/3·9-s − 0.866·12-s + 4.27·14-s + 5/4·16-s − 0.471·18-s + 0.229·19-s + 1.74·21-s + 0.816·24-s − 2·25-s − 0.192·27-s − 4.53·28-s − 2.22·29-s − 1.06·32-s + 1/2·36-s − 0.324·38-s − 1.87·41-s − 2.46·42-s − 1.21·43-s − 0.721·48-s + 34/7·49-s + 2.82·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.286384627590355256677817927703, −7.55115786695816520855805315822, −7.06756557503064388001592546481, −6.68048915127927837981186032967, −6.59547283064248335138694127148, −5.99408682820686103023132488037, −5.52310079772540467707832315072, −5.21083603966480863402338802043, −3.85774434155809618478109209426, −3.58677867797817326821260752741, −3.38136823890594312411975627313, −2.34092707995523044468378525917, −1.95377788914540392982009269037, −0.61960001960809225980571440734, 0,
0.61960001960809225980571440734, 1.95377788914540392982009269037, 2.34092707995523044468378525917, 3.38136823890594312411975627313, 3.58677867797817326821260752741, 3.85774434155809618478109209426, 5.21083603966480863402338802043, 5.52310079772540467707832315072, 5.99408682820686103023132488037, 6.59547283064248335138694127148, 6.68048915127927837981186032967, 7.06756557503064388001592546481, 7.55115786695816520855805315822, 8.286384627590355256677817927703
