| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s − 4·7-s + 4·8-s + 2·10-s + 2·11-s + 2·12-s − 2·13-s − 8·14-s + 15-s + 8·16-s − 2·17-s + 19-s + 2·20-s − 4·21-s + 4·22-s + 4·23-s + 4·24-s − 4·26-s − 27-s − 8·28-s + 5·29-s + 2·30-s + 18·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 1.51·7-s + 1.41·8-s + 0.632·10-s + 0.603·11-s + 0.577·12-s − 0.554·13-s − 2.13·14-s + 0.258·15-s + 2·16-s − 0.485·17-s + 0.229·19-s + 0.447·20-s − 0.872·21-s + 0.852·22-s + 0.834·23-s + 0.816·24-s − 0.784·26-s − 0.192·27-s − 1.51·28-s + 0.928·29-s + 0.365·30-s + 3.23·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.834661579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.834661579\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−12.46209941022812208600273517696, −11.93696717434466127438971427028, −11.21198045734166541035629071582, −10.59004637505799710566969515085, −10.12559445741482174879000747059, −9.910908038325248261984370336318, −9.249830612000781248658361048083, −8.823268492205497817692326779418, −8.149573713843049867930795382755, −7.62215301032654969398137388535, −6.87269569677938404522904086291, −6.44140976779586295962524123115, −6.35392494596635930365160337874, −5.13903439692708512630776699117, −5.11183917912864208719460375644, −4.19230080735664690705498155337, −3.79557217055336642956445750817, −2.81806558401859571988235238245, −2.79162727271473692132161533456, −1.39979474596827242053889525407,
1.39979474596827242053889525407, 2.79162727271473692132161533456, 2.81806558401859571988235238245, 3.79557217055336642956445750817, 4.19230080735664690705498155337, 5.11183917912864208719460375644, 5.13903439692708512630776699117, 6.35392494596635930365160337874, 6.44140976779586295962524123115, 6.87269569677938404522904086291, 7.62215301032654969398137388535, 8.149573713843049867930795382755, 8.823268492205497817692326779418, 9.249830612000781248658361048083, 9.910908038325248261984370336318, 10.12559445741482174879000747059, 10.59004637505799710566969515085, 11.21198045734166541035629071582, 11.93696717434466127438971427028, 12.46209941022812208600273517696
