| L(s) = 1 | + 2-s + 5-s − 3·7-s + 8-s − 9-s + 10-s − 5·11-s − 4·13-s − 3·14-s − 16-s + 4·17-s − 18-s + 19-s − 5·22-s + 2·23-s − 7·25-s − 4·26-s + 5·29-s − 10·31-s − 6·32-s + 4·34-s − 3·35-s − 2·37-s + 38-s + 40-s − 3·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.447·5-s − 1.13·7-s + 0.353·8-s − 1/3·9-s + 0.316·10-s − 1.50·11-s − 1.10·13-s − 0.801·14-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.229·19-s − 1.06·22-s + 0.417·23-s − 7/5·25-s − 0.784·26-s + 0.928·29-s − 1.79·31-s − 1.06·32-s + 0.685·34-s − 0.507·35-s − 0.328·37-s + 0.162·38-s + 0.158·40-s − 0.468·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79857 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79857 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 467 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 27 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 31 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 69 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 76 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 81 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T - 43 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 151 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 173 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.5719507349, −13.7590210186, −13.6835506580, −13.2346874582, −12.7972787778, −12.4451887108, −12.0309448252, −11.4614377492, −10.8260988239, −10.3439984865, −10.0833670404, −9.47761080303, −9.25472981807, −8.42953413197, −7.86570461669, −7.34932638343, −7.02378102715, −6.22361819537, −5.67490490544, −5.19664004453, −4.90150278776, −3.94743889890, −3.32876386306, −2.69937414667, −1.96396109130, 0,
1.96396109130, 2.69937414667, 3.32876386306, 3.94743889890, 4.90150278776, 5.19664004453, 5.67490490544, 6.22361819537, 7.02378102715, 7.34932638343, 7.86570461669, 8.42953413197, 9.25472981807, 9.47761080303, 10.0833670404, 10.3439984865, 10.8260988239, 11.4614377492, 12.0309448252, 12.4451887108, 12.7972787778, 13.2346874582, 13.6835506580, 13.7590210186, 14.5719507349
