| L(s) = 1 | − 2·2-s − 3-s + 2·5-s + 2·6-s − 3·7-s + 4·8-s − 4·9-s − 4·10-s − 3·11-s + 13-s + 6·14-s − 2·15-s − 4·16-s − 17-s + 8·18-s − 19-s + 3·21-s + 6·22-s − 8·23-s − 4·24-s + 2·25-s − 2·26-s + 6·27-s − 3·29-s + 4·30-s − 8·31-s + 3·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 0.894·5-s + 0.816·6-s − 1.13·7-s + 1.41·8-s − 4/3·9-s − 1.26·10-s − 0.904·11-s + 0.277·13-s + 1.60·14-s − 0.516·15-s − 16-s − 0.242·17-s + 1.88·18-s − 0.229·19-s + 0.654·21-s + 1.27·22-s − 1.66·23-s − 0.816·24-s + 2/5·25-s − 0.392·26-s + 1.15·27-s − 0.557·29-s + 0.730·30-s − 1.43·31-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4291 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4291 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 613 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 22 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 39 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9 T + p T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T - 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 96 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 13 T + 82 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 104 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T - 55 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 16 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 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
−17.8263371293, −17.5482464203, −17.2497481295, −16.5423072496, −16.2474811871, −15.8431988047, −14.8013881386, −14.2519380968, −13.7913001558, −13.0685752569, −12.9159994425, −12.0470484289, −11.2160786814, −10.7656253690, −10.1392539013, −9.63596301968, −9.28830758524, −8.53635736947, −8.16779617480, −7.27373807279, −6.23112573409, −5.81848320237, −5.12098675940, −3.78226101860, −2.41780072579, 0,
2.41780072579, 3.78226101860, 5.12098675940, 5.81848320237, 6.23112573409, 7.27373807279, 8.16779617480, 8.53635736947, 9.28830758524, 9.63596301968, 10.1392539013, 10.7656253690, 11.2160786814, 12.0470484289, 12.9159994425, 13.0685752569, 13.7913001558, 14.2519380968, 14.8013881386, 15.8431988047, 16.2474811871, 16.5423072496, 17.2497481295, 17.5482464203, 17.8263371293
