| L(s) = 1 | − 2·3-s − 2·4-s + 3·9-s − 6·11-s + 4·12-s + 4·16-s + 16·17-s − 12·19-s − 9·25-s − 4·27-s + 12·33-s − 6·36-s − 20·41-s − 16·43-s + 12·44-s − 8·48-s − 5·49-s − 32·51-s + 24·57-s − 20·59-s − 8·64-s + 8·67-s − 32·68-s − 16·73-s + 18·75-s + 24·76-s + 5·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 9-s − 1.80·11-s + 1.15·12-s + 16-s + 3.88·17-s − 2.75·19-s − 9/5·25-s − 0.769·27-s + 2.08·33-s − 36-s − 3.12·41-s − 2.43·43-s + 1.80·44-s − 1.15·48-s − 5/7·49-s − 4.48·51-s + 3.17·57-s − 2.60·59-s − 64-s + 0.977·67-s − 3.88·68-s − 1.87·73-s + 2.07·75-s + 2.75·76-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1272384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1272384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−7.51620065265729767387143049492, −7.30736849693508971465380420611, −6.55565745937886916451045260340, −5.89237944027042668230466104811, −5.83486052932247599921983260982, −5.40139681718081078835435283608, −4.78319395015059404410369867924, −4.75349830419579613526576050031, −3.91627384639349895933819843171, −3.31847810114563157193588872502, −3.10754436948669528550295068747, −1.84075228272916268672366687110, −1.45308800162639786710296448498, 0, 0,
1.45308800162639786710296448498, 1.84075228272916268672366687110, 3.10754436948669528550295068747, 3.31847810114563157193588872502, 3.91627384639349895933819843171, 4.75349830419579613526576050031, 4.78319395015059404410369867924, 5.40139681718081078835435283608, 5.83486052932247599921983260982, 5.89237944027042668230466104811, 6.55565745937886916451045260340, 7.30736849693508971465380420611, 7.51620065265729767387143049492
