| L(s) = 1 | − 4-s − 2·9-s − 4·11-s + 2·13-s − 3·16-s + 5·17-s − 4·19-s + 4·23-s + 2·25-s + 2·29-s − 8·31-s + 2·36-s − 2·37-s + 6·41-s + 4·44-s + 8·47-s − 4·49-s − 2·52-s + 8·53-s + 2·61-s + 7·64-s + 4·67-s − 5·68-s − 8·71-s + 14·73-s + 4·76-s − 4·79-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2/3·9-s − 1.20·11-s + 0.554·13-s − 3/4·16-s + 1.21·17-s − 0.917·19-s + 0.834·23-s + 2/5·25-s + 0.371·29-s − 1.43·31-s + 1/3·36-s − 0.328·37-s + 0.937·41-s + 0.603·44-s + 1.16·47-s − 4/7·49-s − 0.277·52-s + 1.09·53-s + 0.256·61-s + 7/8·64-s + 0.488·67-s − 0.606·68-s − 0.949·71-s + 1.63·73-s + 0.458·76-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5395233473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5395233473\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 10 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 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
−19.0459852346, −18.5596302753, −18.1907894208, −17.6220248486, −16.9672695607, −16.4933059333, −15.9226107969, −15.2882381123, −14.6857385787, −14.1241861190, −13.5889051744, −12.8502527551, −12.6156433504, −11.6455338305, −10.9162192013, −10.5851095071, −9.68112699785, −8.91720375300, −8.44325090434, −7.65317943138, −6.81142119319, −5.70285275554, −5.16379094825, −3.99230647147, −2.72641260802,
2.72641260802, 3.99230647147, 5.16379094825, 5.70285275554, 6.81142119319, 7.65317943138, 8.44325090434, 8.91720375300, 9.68112699785, 10.5851095071, 10.9162192013, 11.6455338305, 12.6156433504, 12.8502527551, 13.5889051744, 14.1241861190, 14.6857385787, 15.2882381123, 15.9226107969, 16.4933059333, 16.9672695607, 17.6220248486, 18.1907894208, 18.5596302753, 19.0459852346
