| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 7-s − 4·8-s − 2·9-s − 2·10-s − 6·11-s + 2·14-s + 8·16-s − 3·17-s + 4·18-s − 3·19-s + 2·20-s + 12·22-s − 3·23-s − 2·25-s − 3·27-s − 2·28-s − 3·29-s − 8·32-s + 6·34-s − 35-s − 4·36-s + 10·37-s + 6·38-s − 4·40-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s − 1.41·8-s − 2/3·9-s − 0.632·10-s − 1.80·11-s + 0.534·14-s + 2·16-s − 0.727·17-s + 0.942·18-s − 0.688·19-s + 0.447·20-s + 2.55·22-s − 0.625·23-s − 2/5·25-s − 0.577·27-s − 0.377·28-s − 0.557·29-s − 1.41·32-s + 1.02·34-s − 0.169·35-s − 2/3·36-s + 1.64·37-s + 0.973·38-s − 0.632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10023 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10023 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 257 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 24 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T - 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - T + 31 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 42 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 135 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 93 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 145 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 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
−16.9315675446, −16.4103256545, −16.0611751297, −15.4102538548, −15.0152858908, −14.6487150518, −13.7251714102, −13.2674183342, −12.8971819445, −12.2149951824, −11.6369800362, −10.9789253317, −10.6000656074, −9.90791884600, −9.64650193093, −8.87851560574, −8.61161327080, −7.79606558193, −7.55809667704, −6.42905021193, −5.89625010754, −5.46682937590, −4.20503897888, −2.90536555045, −2.31894101908, 0,
2.31894101908, 2.90536555045, 4.20503897888, 5.46682937590, 5.89625010754, 6.42905021193, 7.55809667704, 7.79606558193, 8.61161327080, 8.87851560574, 9.64650193093, 9.90791884600, 10.6000656074, 10.9789253317, 11.6369800362, 12.2149951824, 12.8971819445, 13.2674183342, 13.7251714102, 14.6487150518, 15.0152858908, 15.4102538548, 16.0611751297, 16.4103256545, 16.9315675446
