| L(s) = 1 | − 2·2-s − 3·3-s + 6·6-s − 6·7-s + 4·8-s + 4·9-s + 4·11-s − 2·13-s + 12·14-s − 4·16-s − 4·17-s − 8·18-s + 18·21-s − 8·22-s − 4·23-s − 12·24-s − 25-s + 4·26-s + 4·29-s − 10·31-s − 12·33-s + 8·34-s + 2·37-s + 6·39-s − 6·41-s − 36·42-s − 2·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 2.44·6-s − 2.26·7-s + 1.41·8-s + 4/3·9-s + 1.20·11-s − 0.554·13-s + 3.20·14-s − 16-s − 0.970·17-s − 1.88·18-s + 3.92·21-s − 1.70·22-s − 0.834·23-s − 2.44·24-s − 1/5·25-s + 0.784·26-s + 0.742·29-s − 1.79·31-s − 2.08·33-s + 1.37·34-s + 0.328·37-s + 0.960·39-s − 0.937·41-s − 5.55·42-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−19.6970630576, −19.5490432498, −18.9344925121, −18.1213987035, −18.0285451209, −17.4264315599, −16.7022471733, −16.5806200716, −16.2562118377, −15.3791583737, −14.5011418501, −13.3514918073, −13.2593691405, −12.1048029115, −12.0629600164, −11.0017900091, −10.2432044592, −9.91407057019, −9.18016854907, −8.83642738145, −7.47848778675, −6.41173993137, −6.39084306103, −5.03912355415, −3.81591641511, 0,
3.81591641511, 5.03912355415, 6.39084306103, 6.41173993137, 7.47848778675, 8.83642738145, 9.18016854907, 9.91407057019, 10.2432044592, 11.0017900091, 12.0629600164, 12.1048029115, 13.2593691405, 13.3514918073, 14.5011418501, 15.3791583737, 16.2562118377, 16.5806200716, 16.7022471733, 17.4264315599, 18.0285451209, 18.1213987035, 18.9344925121, 19.5490432498, 19.6970630576
