| L(s) = 1 | + 3-s + 4-s − 5-s − 2·9-s + 4·11-s + 12-s − 3·13-s − 15-s − 3·16-s − 2·17-s − 20-s + 6·23-s − 2·25-s − 2·27-s − 3·29-s − 8·31-s + 4·33-s − 2·36-s − 3·39-s + 10·41-s − 4·43-s + 4·44-s + 2·45-s + 8·47-s − 3·48-s + 2·49-s − 2·51-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.447·5-s − 2/3·9-s + 1.20·11-s + 0.288·12-s − 0.832·13-s − 0.258·15-s − 3/4·16-s − 0.485·17-s − 0.223·20-s + 1.25·23-s − 2/5·25-s − 0.384·27-s − 0.557·29-s − 1.43·31-s + 0.696·33-s − 1/3·36-s − 0.480·39-s + 1.56·41-s − 0.609·43-s + 0.603·44-s + 0.298·45-s + 1.16·47-s − 0.433·48-s + 2/7·49-s − 0.280·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5655 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5655 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.003884372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003884372\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−17.3048992634, −16.7458072539, −16.4256204677, −15.7719544743, −15.0998635006, −14.8471809880, −14.4563362575, −13.7213753796, −13.3873264422, −12.5010838871, −12.0787147398, −11.5493513634, −10.9784942142, −10.6461337301, −9.45666758574, −9.06207918836, −8.85263161912, −7.65778341205, −7.42400991016, −6.62178494498, −5.94617751973, −4.92180155854, −4.06774477771, −3.14844113553, −2.15160002299,
2.15160002299, 3.14844113553, 4.06774477771, 4.92180155854, 5.94617751973, 6.62178494498, 7.42400991016, 7.65778341205, 8.85263161912, 9.06207918836, 9.45666758574, 10.6461337301, 10.9784942142, 11.5493513634, 12.0787147398, 12.5010838871, 13.3873264422, 13.7213753796, 14.4563362575, 14.8471809880, 15.0998635006, 15.7719544743, 16.4256204677, 16.7458072539, 17.3048992634
