| L(s) = 1 | − 3-s − 2·4-s − 5-s − 2·7-s + 3·9-s + 11-s + 2·12-s − 2·13-s + 15-s + 4·16-s + 2·20-s + 2·21-s + 3·23-s − 3·25-s − 8·27-s + 4·28-s + 2·29-s + 3·31-s − 33-s + 2·35-s − 6·36-s − 7·37-s + 2·39-s + 2·41-s + 2·43-s − 2·44-s − 3·45-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 0.755·7-s + 9-s + 0.301·11-s + 0.577·12-s − 0.554·13-s + 0.258·15-s + 16-s + 0.447·20-s + 0.436·21-s + 0.625·23-s − 3/5·25-s − 1.53·27-s + 0.755·28-s + 0.371·29-s + 0.538·31-s − 0.174·33-s + 0.338·35-s − 36-s − 1.15·37-s + 0.320·39-s + 0.312·41-s + 0.304·43-s − 0.301·44-s − 0.447·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3326674195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3326674195\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 13 T + 142 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T + 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 136 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 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
−19.7222594961, −19.2944286986, −19.1240581393, −18.2633049007, −17.8210350828, −17.1589973209, −16.7082306195, −15.9937244189, −15.4004182952, −14.8307008242, −13.9840452384, −13.2948510086, −12.8466456083, −12.1534411366, −11.6039745628, −10.5503901602, −9.89613033703, −9.40678387434, −8.47300195185, −7.51220016117, −6.72526529187, −5.61896007956, −4.61175629643, −3.65534021907,
3.65534021907, 4.61175629643, 5.61896007956, 6.72526529187, 7.51220016117, 8.47300195185, 9.40678387434, 9.89613033703, 10.5503901602, 11.6039745628, 12.1534411366, 12.8466456083, 13.2948510086, 13.9840452384, 14.8307008242, 15.4004182952, 15.9937244189, 16.7082306195, 17.1589973209, 17.8210350828, 18.2633049007, 19.1240581393, 19.2944286986, 19.7222594961
