| L(s) = 1 | − 2·2-s − 3·3-s − 5·5-s + 6·6-s − 5·7-s + 4·8-s + 2·9-s + 10·10-s − 2·11-s − 5·13-s + 10·14-s + 15·15-s − 4·16-s − 6·17-s − 4·18-s − 2·19-s + 15·21-s + 4·22-s − 11·23-s − 12·24-s + 10·25-s + 10·26-s + 6·27-s − 30·30-s − 4·31-s + 6·33-s + 12·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s − 2.23·5-s + 2.44·6-s − 1.88·7-s + 1.41·8-s + 2/3·9-s + 3.16·10-s − 0.603·11-s − 1.38·13-s + 2.67·14-s + 3.87·15-s − 16-s − 1.45·17-s − 0.942·18-s − 0.458·19-s + 3.27·21-s + 0.852·22-s − 2.29·23-s − 2.44·24-s + 2·25-s + 1.96·26-s + 1.15·27-s − 5.47·30-s − 0.718·31-s + 1.04·33-s + 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65869 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65869 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 199 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 24 T + p T^{2} ) \) |
| 331 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 4 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 13 T + 92 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 106 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 91 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
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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
−15.4760314332, −15.0336665168, −14.2680185140, −13.6249490424, −13.1885665860, −12.5309515367, −12.3106161086, −11.8619126970, −11.5834481999, −11.0093216697, −10.4348397042, −10.2448713868, −9.66874245173, −9.19437141573, −8.57091612661, −8.14990029344, −7.73829884693, −7.07409884500, −6.70725340691, −6.08646651846, −5.38816255787, −4.72065211272, −4.11522595370, −3.65354964410, −2.52973166474, 0, 0, 0,
2.52973166474, 3.65354964410, 4.11522595370, 4.72065211272, 5.38816255787, 6.08646651846, 6.70725340691, 7.07409884500, 7.73829884693, 8.14990029344, 8.57091612661, 9.19437141573, 9.66874245173, 10.2448713868, 10.4348397042, 11.0093216697, 11.5834481999, 11.8619126970, 12.3106161086, 12.5309515367, 13.1885665860, 13.6249490424, 14.2680185140, 15.0336665168, 15.4760314332
