| L(s) = 1 | − 2·2-s − 4·3-s − 5-s + 8·6-s − 4·7-s + 4·8-s + 7·9-s + 2·10-s − 10·11-s + 2·13-s + 8·14-s + 4·15-s − 4·16-s − 4·17-s − 14·18-s − 8·19-s + 16·21-s + 20·22-s + 6·23-s − 16·24-s − 2·25-s − 4·26-s − 4·27-s + 10·29-s − 8·30-s − 2·31-s + 40·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s − 0.447·5-s + 3.26·6-s − 1.51·7-s + 1.41·8-s + 7/3·9-s + 0.632·10-s − 3.01·11-s + 0.554·13-s + 2.13·14-s + 1.03·15-s − 16-s − 0.970·17-s − 3.29·18-s − 1.83·19-s + 3.49·21-s + 4.26·22-s + 1.25·23-s − 3.26·24-s − 2/5·25-s − 0.784·26-s − 0.769·27-s + 1.85·29-s − 1.46·30-s − 0.359·31-s + 6.96·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 37 | $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$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
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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
−17.5309872372, −17.1416936480, −16.8856657579, −16.1920174169, −15.8928365109, −15.6038578732, −14.9798366844, −13.5160814577, −13.2856056386, −12.9583864139, −12.3936029592, −11.7573247228, −10.8216312508, −10.7751381625, −10.3300470506, −9.93309835361, −8.99824486387, −8.37102641740, −8.01433080787, −6.87039121695, −6.53964005794, −5.74480366917, −5.00317001401, −4.56875092817, −2.94510070475, 0, 0,
2.94510070475, 4.56875092817, 5.00317001401, 5.74480366917, 6.53964005794, 6.87039121695, 8.01433080787, 8.37102641740, 8.99824486387, 9.93309835361, 10.3300470506, 10.7751381625, 10.8216312508, 11.7573247228, 12.3936029592, 12.9583864139, 13.2856056386, 13.5160814577, 14.9798366844, 15.6038578732, 15.8928365109, 16.1920174169, 16.8856657579, 17.1416936480, 17.5309872372
