| L(s) = 1 | − 2·2-s − 2·3-s + 4-s − 5-s + 4·6-s − 3·7-s + 2·9-s + 2·10-s − 3·11-s − 2·12-s + 13-s + 6·14-s + 2·15-s + 16-s − 6·17-s − 4·18-s − 20-s + 6·21-s + 6·22-s + 23-s − 5·25-s − 2·26-s − 6·27-s − 3·28-s − 7·29-s − 4·30-s + 31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s + 1.63·6-s − 1.13·7-s + 2/3·9-s + 0.632·10-s − 0.904·11-s − 0.577·12-s + 0.277·13-s + 1.60·14-s + 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.942·18-s − 0.223·20-s + 1.30·21-s + 1.27·22-s + 0.208·23-s − 25-s − 0.392·26-s − 1.15·27-s − 0.566·28-s − 1.29·29-s − 0.730·30-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64577 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64577 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 64577 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 218 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 49 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T - 29 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 49 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 39 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 15 T + 120 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 55 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 129 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 13 T + 179 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−15.0884401877, −14.8582524880, −13.8134929571, −13.3122984401, −13.1204103259, −12.7635950224, −12.0181898719, −11.5484405511, −11.3198806548, −10.7620639039, −10.2860353967, −9.83784393078, −9.40929317518, −9.08602066405, −8.42143605912, −7.92553106328, −7.48846996650, −6.81354746168, −6.22782957431, −5.96287530187, −5.14212843836, −4.53461234336, −3.72419886705, −2.97340615460, −1.78415551534, 0, 0,
1.78415551534, 2.97340615460, 3.72419886705, 4.53461234336, 5.14212843836, 5.96287530187, 6.22782957431, 6.81354746168, 7.48846996650, 7.92553106328, 8.42143605912, 9.08602066405, 9.40929317518, 9.83784393078, 10.2860353967, 10.7620639039, 11.3198806548, 11.5484405511, 12.0181898719, 12.7635950224, 13.1204103259, 13.3122984401, 13.8134929571, 14.8582524880, 15.0884401877
