| L(s) = 1 | − 2·2-s − 4·3-s + 8·6-s − 3·7-s + 4·8-s + 7·9-s + 6·14-s − 4·16-s − 14·18-s − 6·19-s + 12·21-s − 6·23-s − 16·24-s − 9·25-s − 4·27-s + 10·29-s + 8·31-s − 2·37-s + 12·38-s − 10·41-s − 24·42-s − 14·43-s + 12·46-s + 16·47-s + 16·48-s + 18·50-s − 12·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 3.26·6-s − 1.13·7-s + 1.41·8-s + 7/3·9-s + 1.60·14-s − 16-s − 3.29·18-s − 1.37·19-s + 2.61·21-s − 1.25·23-s − 3.26·24-s − 9/5·25-s − 0.769·27-s + 1.85·29-s + 1.43·31-s − 0.328·37-s + 1.94·38-s − 1.56·41-s − 3.70·42-s − 2.13·43-s + 1.76·46-s + 2.33·47-s + 2.30·48-s + 2.54·50-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−19.5612669259, −19.1857249719, −18.7959296475, −17.9414335735, −17.7484336353, −17.1998451979, −17.0336103204, −16.2849525785, −15.9140726033, −15.2043108428, −13.7579805783, −13.5686390571, −12.4153807038, −12.0957944883, −11.4512586103, −10.5446227328, −10.0355090972, −9.84275137950, −8.60353961929, −8.09012557482, −6.56709274806, −6.36261389471, −5.30170245061, −4.26020787259, 0,
4.26020787259, 5.30170245061, 6.36261389471, 6.56709274806, 8.09012557482, 8.60353961929, 9.84275137950, 10.0355090972, 10.5446227328, 11.4512586103, 12.0957944883, 12.4153807038, 13.5686390571, 13.7579805783, 15.2043108428, 15.9140726033, 16.2849525785, 17.0336103204, 17.1998451979, 17.7484336353, 17.9414335735, 18.7959296475, 19.1857249719, 19.5612669259
