| L(s) = 1 | − 2·2-s − 3·3-s − 4·5-s + 6·6-s − 4·7-s + 4·8-s + 2·9-s + 8·10-s − 11-s − 2·13-s + 8·14-s + 12·15-s − 4·16-s − 17-s − 4·18-s − 4·19-s + 12·21-s + 2·22-s − 2·23-s − 12·24-s + 6·25-s + 4·26-s + 6·27-s − 24·30-s − 8·31-s + 3·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s − 1.78·5-s + 2.44·6-s − 1.51·7-s + 1.41·8-s + 2/3·9-s + 2.52·10-s − 0.301·11-s − 0.554·13-s + 2.13·14-s + 3.09·15-s − 16-s − 0.242·17-s − 0.942·18-s − 0.917·19-s + 2.61·21-s + 0.426·22-s − 0.417·23-s − 2.44·24-s + 6/5·25-s + 0.784·26-s + 1.15·27-s − 4.38·30-s − 1.43·31-s + 0.522·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6443 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6443 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 379 | $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 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T - 54 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 48 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 21 T + 200 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T - 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 196 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 47 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 99 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T - 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 13 T + 190 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.6383823334, −17.0437609635, −16.8053996113, −16.4770379760, −15.8918727498, −15.5790457738, −14.8987043261, −14.1629337721, −13.3340821798, −12.6641468560, −12.4412043327, −11.7997319363, −11.2686447720, −10.8102494191, −10.3188853638, −9.67058950569, −9.06814272994, −8.55511225271, −7.71394550514, −7.44703411728, −6.40798053029, −5.99473466071, −4.88901509833, −4.28946492863, −3.26272611850, 0, 0,
3.26272611850, 4.28946492863, 4.88901509833, 5.99473466071, 6.40798053029, 7.44703411728, 7.71394550514, 8.55511225271, 9.06814272994, 9.67058950569, 10.3188853638, 10.8102494191, 11.2686447720, 11.7997319363, 12.4412043327, 12.6641468560, 13.3340821798, 14.1629337721, 14.8987043261, 15.5790457738, 15.8918727498, 16.4770379760, 16.8053996113, 17.0437609635, 17.6383823334
