| L(s) = 1 | − 3-s + 4-s − 2·5-s − 7-s − 3·9-s − 2·11-s − 12-s + 6·13-s + 2·15-s + 16-s − 3·19-s − 2·20-s + 21-s + 2·23-s + 2·25-s + 4·27-s − 28-s + 29-s − 4·31-s + 2·33-s + 2·35-s − 3·36-s + 2·37-s − 6·39-s − 3·41-s + 6·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.377·7-s − 9-s − 0.603·11-s − 0.288·12-s + 1.66·13-s + 0.516·15-s + 1/4·16-s − 0.688·19-s − 0.447·20-s + 0.218·21-s + 0.417·23-s + 2/5·25-s + 0.769·27-s − 0.188·28-s + 0.185·29-s − 0.718·31-s + 0.348·33-s + 0.338·35-s − 1/2·36-s + 0.328·37-s − 0.960·39-s − 0.468·41-s + 0.914·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1180 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1180 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4476020707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4476020707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 12 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 146 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 34 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 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
−19.7422974351, −19.1201102685, −18.7329955429, −18.0513020875, −17.4986983936, −16.7475467493, −16.3722456827, −15.9151042999, −15.2177860643, −14.9192462121, −13.8593789105, −13.4274715487, −12.5326334002, −12.0918100119, −11.3268599930, −10.8531808931, −10.5888703293, −9.23482442020, −8.56208749819, −7.92658426890, −6.99620405404, −6.11622298470, −5.54774220450, −4.13792671166, −3.01588656316,
3.01588656316, 4.13792671166, 5.54774220450, 6.11622298470, 6.99620405404, 7.92658426890, 8.56208749819, 9.23482442020, 10.5888703293, 10.8531808931, 11.3268599930, 12.0918100119, 12.5326334002, 13.4274715487, 13.8593789105, 14.9192462121, 15.2177860643, 15.9151042999, 16.3722456827, 16.7475467493, 17.4986983936, 18.0513020875, 18.7329955429, 19.1201102685, 19.7422974351
