| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s − 7-s + 8-s − 9-s − 2·11-s + 12-s + 13-s + 14-s − 16-s + 17-s + 18-s + 3·19-s + 21-s + 2·22-s − 23-s − 24-s − 26-s + 28-s + 29-s + 11·31-s + 5·32-s + 2·33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 1/3·9-s − 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.688·19-s + 0.218·21-s + 0.426·22-s − 0.208·23-s − 0.204·24-s − 0.196·26-s + 0.188·28-s + 0.185·29-s + 1.97·31-s + 0.883·32-s + 0.348·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80533 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80533 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 2777 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 101 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 63 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 5 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T - 71 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 80 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 22 T + 244 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 21 T + 225 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 128 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 224 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−14.3356658209, −14.1145204352, −13.5327085459, −13.1159498453, −12.8344660546, −12.1105389265, −11.7365589091, −11.4416691526, −10.8255743578, −10.2628644828, −9.94582852513, −9.59448235994, −8.88501303381, −8.62750297703, −8.08873854721, −7.58206604759, −6.96695262100, −6.38157569306, −5.71699720371, −5.49696069861, −4.55179258187, −4.24066661773, −3.12695068512, −2.65138532218, −1.20360306027, 0,
1.20360306027, 2.65138532218, 3.12695068512, 4.24066661773, 4.55179258187, 5.49696069861, 5.71699720371, 6.38157569306, 6.96695262100, 7.58206604759, 8.08873854721, 8.62750297703, 8.88501303381, 9.59448235994, 9.94582852513, 10.2628644828, 10.8255743578, 11.4416691526, 11.7365589091, 12.1105389265, 12.8344660546, 13.1159498453, 13.5327085459, 14.1145204352, 14.3356658209
