| L(s) = 1 | − 2·2-s + 3·3-s + 2·5-s − 6·6-s + 2·7-s + 4·8-s + 2·9-s − 4·10-s − 3·11-s − 2·13-s − 4·14-s + 6·15-s − 4·16-s − 4·18-s − 2·19-s + 6·21-s + 6·22-s + 3·23-s + 12·24-s − 2·25-s + 4·26-s − 6·27-s − 6·29-s − 12·30-s + 2·31-s − 9·33-s + 4·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 0.894·5-s − 2.44·6-s + 0.755·7-s + 1.41·8-s + 2/3·9-s − 1.26·10-s − 0.904·11-s − 0.554·13-s − 1.06·14-s + 1.54·15-s − 16-s − 0.942·18-s − 0.458·19-s + 1.30·21-s + 1.27·22-s + 0.625·23-s + 2.44·24-s − 2/5·25-s + 0.784·26-s − 1.15·27-s − 1.11·29-s − 2.19·30-s + 0.359·31-s − 1.56·33-s + 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4571 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4571 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6349912032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6349912032\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 653 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 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 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 100 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T - 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5 T - 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−17.6315654406, −17.4269680734, −16.9061813861, −16.2934769230, −15.4131726406, −14.9510863599, −14.4820810703, −13.9556026704, −13.7089857730, −13.0652593545, −12.6447024605, −11.4591060153, −10.9262217642, −10.1089537988, −9.79595487230, −9.13748871302, −8.83301993654, −8.33861129542, −7.70930887671, −7.44014604627, −5.96170776823, −5.14664583502, −4.18793618209, −2.84587673646, −2.00326422459,
2.00326422459, 2.84587673646, 4.18793618209, 5.14664583502, 5.96170776823, 7.44014604627, 7.70930887671, 8.33861129542, 8.83301993654, 9.13748871302, 9.79595487230, 10.1089537988, 10.9262217642, 11.4591060153, 12.6447024605, 13.0652593545, 13.7089857730, 13.9556026704, 14.4820810703, 14.9510863599, 15.4131726406, 16.2934769230, 16.9061813861, 17.4269680734, 17.6315654406
