| L(s) = 1 | − 2·2-s − 3-s − 2·5-s + 2·6-s − 7-s + 4·8-s − 2·9-s + 4·10-s − 2·11-s − 6·13-s + 2·14-s + 2·15-s − 4·16-s + 17-s + 4·18-s − 3·19-s + 21-s + 4·22-s + 8·23-s − 4·24-s + 2·25-s + 12·26-s + 2·27-s + 29-s − 4·30-s − 31-s + 2·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s − 0.894·5-s + 0.816·6-s − 0.377·7-s + 1.41·8-s − 2/3·9-s + 1.26·10-s − 0.603·11-s − 1.66·13-s + 0.534·14-s + 0.516·15-s − 16-s + 0.242·17-s + 0.942·18-s − 0.688·19-s + 0.218·21-s + 0.852·22-s + 1.66·23-s − 0.816·24-s + 2/5·25-s + 2.35·26-s + 0.384·27-s + 0.185·29-s − 0.730·30-s − 0.179·31-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2101 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2101 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 191 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 10 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - T + 35 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 76 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T + 74 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T - 15 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 5 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−18.9955139191, −18.5422530578, −17.9573639826, −17.3598344955, −17.0955615533, −16.6849773552, −16.1266990140, −15.3752472468, −14.7448497084, −14.3689001295, −13.3693186613, −12.7771707590, −12.3495163538, −11.4524787183, −11.0419580693, −10.3442115301, −9.64581787199, −9.25181112947, −8.42997195515, −7.90462179048, −7.32978371723, −6.37957734098, −5.10491705854, −4.62203036355, −3.03769879901, 0,
3.03769879901, 4.62203036355, 5.10491705854, 6.37957734098, 7.32978371723, 7.90462179048, 8.42997195515, 9.25181112947, 9.64581787199, 10.3442115301, 11.0419580693, 11.4524787183, 12.3495163538, 12.7771707590, 13.3693186613, 14.3689001295, 14.7448497084, 15.3752472468, 16.1266990140, 16.6849773552, 17.0955615533, 17.3598344955, 17.9573639826, 18.5422530578, 18.9955139191
