| L(s) = 1 | − 2-s − 6·3-s − 4-s + 6·6-s + 3·8-s + 21·9-s + 6·12-s − 16-s − 6·17-s − 21·18-s − 10·19-s − 18·24-s − 10·25-s − 54·27-s − 5·32-s + 6·34-s − 21·36-s + 10·38-s + 12·41-s − 4·43-s + 6·48-s + 2·49-s + 10·50-s + 36·51-s + 54·54-s + 60·57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 3.46·3-s − 1/2·4-s + 2.44·6-s + 1.06·8-s + 7·9-s + 1.73·12-s − 1/4·16-s − 1.45·17-s − 4.94·18-s − 2.29·19-s − 3.67·24-s − 2·25-s − 10.3·27-s − 0.883·32-s + 1.02·34-s − 7/2·36-s + 1.62·38-s + 1.87·41-s − 0.609·43-s + 0.866·48-s + 2/7·49-s + 1.41·50-s + 5.04·51-s + 7.34·54-s + 7.94·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−8.820142471571876337599747567018, −8.205174103101839964623123933331, −7.40585307678501251492741626335, −7.20736947999270954805514319970, −6.49177047979490854052512324082, −6.04347889941024738587947019148, −5.95808888910984261709863888066, −5.27863549534731961075900677576, −4.50862835068296220400083952922, −4.42654752979487491573226444690, −4.04318522706247983721491253685, −2.12271179066542804694951246257, −1.33598343949561168216183034148, 0, 0,
1.33598343949561168216183034148, 2.12271179066542804694951246257, 4.04318522706247983721491253685, 4.42654752979487491573226444690, 4.50862835068296220400083952922, 5.27863549534731961075900677576, 5.95808888910984261709863888066, 6.04347889941024738587947019148, 6.49177047979490854052512324082, 7.20736947999270954805514319970, 7.40585307678501251492741626335, 8.205174103101839964623123933331, 8.820142471571876337599747567018
