| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s − 5-s + 6·6-s − 4·7-s − 3·8-s + 9-s + 3·10-s − 5·11-s − 8·12-s + 13-s + 12·14-s + 2·15-s + 3·16-s + 17-s − 3·18-s + 3·19-s − 4·20-s + 8·21-s + 15·22-s − 2·23-s + 6·24-s + 25-s − 3·26-s − 2·27-s − 16·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s − 0.447·5-s + 2.44·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.948·10-s − 1.50·11-s − 2.30·12-s + 0.277·13-s + 3.20·14-s + 0.516·15-s + 3/4·16-s + 0.242·17-s − 0.707·18-s + 0.688·19-s − 0.894·20-s + 1.74·21-s + 3.19·22-s − 0.417·23-s + 1.22·24-s + 1/5·25-s − 0.588·26-s − 0.384·27-s − 3.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 48 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 110 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 88 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 16 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 94 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T - 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 13 T + 150 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 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
−19.7797425606, −19.1978831065, −18.6560008587, −18.5769436298, −17.7446580712, −17.5800183591, −16.7052225316, −16.4960124946, −15.8590954838, −15.6288915940, −14.6469159194, −13.3600382646, −13.0913401763, −12.1433305996, −11.6714788680, −10.8443246732, −10.2065432488, −9.93180814793, −9.21023182930, −8.35447984205, −7.78385567028, −6.94600338150, −6.05749108425, −5.22251984953, −3.29054256031, 0,
3.29054256031, 5.22251984953, 6.05749108425, 6.94600338150, 7.78385567028, 8.35447984205, 9.21023182930, 9.93180814793, 10.2065432488, 10.8443246732, 11.6714788680, 12.1433305996, 13.0913401763, 13.3600382646, 14.6469159194, 15.6288915940, 15.8590954838, 16.4960124946, 16.7052225316, 17.5800183591, 17.7446580712, 18.5769436298, 18.6560008587, 19.1978831065, 19.7797425606
