| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s + 5-s + 6·6-s − 7-s − 4·8-s + 6·9-s − 2·10-s − 3·11-s − 6·12-s − 6·13-s + 2·14-s − 3·15-s + 8·16-s − 4·17-s − 12·18-s − 4·19-s + 2·20-s + 3·21-s + 6·22-s − 4·23-s + 12·24-s + 12·26-s − 9·27-s − 2·28-s + 29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s + 0.447·5-s + 2.44·6-s − 0.377·7-s − 1.41·8-s + 2·9-s − 0.632·10-s − 0.904·11-s − 1.73·12-s − 1.66·13-s + 0.534·14-s − 0.774·15-s + 2·16-s − 0.970·17-s − 2.82·18-s − 0.917·19-s + 0.447·20-s + 0.654·21-s + 1.27·22-s − 0.834·23-s + 2.44·24-s + 2.35·26-s − 1.73·27-s − 0.377·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 13 T + 86 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−11.28790245124406219325834414461, −10.85599475480889022452999525775, −10.28100658031774493075400371767, −10.23381064561320180503101258441, −9.558411675696091668225550396574, −9.327142254780200113141852011914, −8.757670347959424035274928629413, −8.083729258950178862141271812890, −7.55761457986048697693512678669, −7.08062239230232568816955140112, −6.56357979723283916516423988025, −5.91938480869255224945259338172, −5.84238684194872165135560994307, −4.86355685640166565028373406273, −4.68138555273270764845834952430, −3.47942312485617092946418405229, −2.49789149299085538948674805055, −1.77021526492042973258993541278, 0, 0,
1.77021526492042973258993541278, 2.49789149299085538948674805055, 3.47942312485617092946418405229, 4.68138555273270764845834952430, 4.86355685640166565028373406273, 5.84238684194872165135560994307, 5.91938480869255224945259338172, 6.56357979723283916516423988025, 7.08062239230232568816955140112, 7.55761457986048697693512678669, 8.083729258950178862141271812890, 8.757670347959424035274928629413, 9.327142254780200113141852011914, 9.558411675696091668225550396574, 10.23381064561320180503101258441, 10.28100658031774493075400371767, 10.85599475480889022452999525775, 11.28790245124406219325834414461
