| L(s) = 1 | + 3-s + 5-s − 4·7-s + 3·9-s + 5·11-s − 4·13-s + 15-s + 3·17-s + 3·19-s − 4·21-s + 23-s + 5·25-s + 8·27-s − 20·29-s + 11·31-s + 5·33-s − 4·35-s − 7·37-s − 4·39-s − 2·41-s + 16·43-s + 3·45-s + 7·47-s + 9·49-s + 3·51-s + 11·53-s + 5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 9-s + 1.50·11-s − 1.10·13-s + 0.258·15-s + 0.727·17-s + 0.688·19-s − 0.872·21-s + 0.208·23-s + 25-s + 1.53·27-s − 3.71·29-s + 1.97·31-s + 0.870·33-s − 0.676·35-s − 1.15·37-s − 0.640·39-s − 0.312·41-s + 2.43·43-s + 0.447·45-s + 1.02·47-s + 9/7·49-s + 0.420·51-s + 1.51·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.916922122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.916922122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 5 T - 48 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.843194629798360725010746390678, −9.664303771330357933669447041705, −9.133155327122756312966308034976, −8.866689842337609554039035466799, −8.797733400188825772003259141151, −7.71631351194631518032281633100, −7.30529306012363243290356955598, −7.29178221018063956457510863140, −6.76578586891479422091967928991, −6.33134364212133292015079266860, −5.66156370713247587826042427706, −5.60436204347377688606277733808, −4.74424878987062399324459939309, −4.20603108264576271409834714921, −3.83094836733668712886190321263, −3.30585998392088565279308606388, −2.82829821525619053579961886191, −2.26879067946146416498864663867, −1.46761439352581439677629190486, −0.77109845396288235273662873895,
0.77109845396288235273662873895, 1.46761439352581439677629190486, 2.26879067946146416498864663867, 2.82829821525619053579961886191, 3.30585998392088565279308606388, 3.83094836733668712886190321263, 4.20603108264576271409834714921, 4.74424878987062399324459939309, 5.60436204347377688606277733808, 5.66156370713247587826042427706, 6.33134364212133292015079266860, 6.76578586891479422091967928991, 7.29178221018063956457510863140, 7.30529306012363243290356955598, 7.71631351194631518032281633100, 8.797733400188825772003259141151, 8.866689842337609554039035466799, 9.133155327122756312966308034976, 9.664303771330357933669447041705, 9.843194629798360725010746390678
