| L(s) = 1 | − 2·2-s + 3-s − 4-s − 5-s − 2·6-s + 8·8-s + 9-s + 2·10-s − 12-s + 4·13-s − 15-s − 7·16-s − 2·18-s + 20-s + 8·24-s + 25-s − 8·26-s + 27-s + 2·30-s − 14·32-s − 36-s + 4·39-s − 8·40-s − 20·41-s − 45-s − 7·48-s − 7·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s + 2.82·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s + 1.10·13-s − 0.258·15-s − 7/4·16-s − 0.471·18-s + 0.223·20-s + 1.63·24-s + 1/5·25-s − 1.56·26-s + 0.192·27-s + 0.365·30-s − 2.47·32-s − 1/6·36-s + 0.640·39-s − 1.26·40-s − 3.12·41-s − 0.149·45-s − 1.01·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.838960281218345298135626667274, −8.455181921164028207665051683361, −8.413264856684473334036504393340, −7.68993943182315499613922198544, −7.46261028773657489832215471574, −6.74126848758315603293550810292, −6.20681526904542088835730053039, −5.28294324247232318062401992920, −4.79492945983632365647604814265, −4.36459226831434420870816413097, −3.55714635244582915245852508954, −3.30593125964790138739196334719, −1.84339419921478037654642459241, −1.27025970316177507740293768624, 0,
1.27025970316177507740293768624, 1.84339419921478037654642459241, 3.30593125964790138739196334719, 3.55714635244582915245852508954, 4.36459226831434420870816413097, 4.79492945983632365647604814265, 5.28294324247232318062401992920, 6.20681526904542088835730053039, 6.74126848758315603293550810292, 7.46261028773657489832215471574, 7.68993943182315499613922198544, 8.413264856684473334036504393340, 8.455181921164028207665051683361, 8.838960281218345298135626667274
