| L(s) = 1 | − 2·3-s + 3·9-s + 4·11-s − 2·13-s + 4·17-s + 8·23-s − 2·25-s − 4·27-s + 4·29-s + 8·31-s − 8·33-s − 4·37-s + 4·39-s + 16·41-s − 8·43-s + 12·47-s − 6·49-s − 8·51-s − 4·53-s − 4·59-s + 4·61-s − 8·67-s − 16·69-s − 4·71-s + 12·73-s + 4·75-s + 5·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 1.20·11-s − 0.554·13-s + 0.970·17-s + 1.66·23-s − 2/5·25-s − 0.769·27-s + 0.742·29-s + 1.43·31-s − 1.39·33-s − 0.657·37-s + 0.640·39-s + 2.49·41-s − 1.21·43-s + 1.75·47-s − 6/7·49-s − 1.12·51-s − 0.549·53-s − 0.520·59-s + 0.512·61-s − 0.977·67-s − 1.92·69-s − 0.474·71-s + 1.40·73-s + 0.461·75-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.463222375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.463222375\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.70416955903848286505051234552, −10.62477045879607834404759818228, −9.823494486222086203985184051987, −9.775061664743793348863091123208, −9.003827506150334562556987410073, −8.953040170137146208791208275335, −8.026354800464021329450976281226, −7.72109461726772599695639038795, −7.15809059112063480398815568835, −6.77040452003920678553009328401, −6.13401519988824419414369840336, −6.13358291589692035874291008449, −5.10212935991382657628656380930, −5.08547849075077576834357829006, −4.36517567221026176113358586257, −3.86923908462354940211166291468, −3.14751495423018683417630639054, −2.47136216059574677501867764567, −1.36526861496787083794537444794, −0.828386562033167066759960164945,
0.828386562033167066759960164945, 1.36526861496787083794537444794, 2.47136216059574677501867764567, 3.14751495423018683417630639054, 3.86923908462354940211166291468, 4.36517567221026176113358586257, 5.08547849075077576834357829006, 5.10212935991382657628656380930, 6.13358291589692035874291008449, 6.13401519988824419414369840336, 6.77040452003920678553009328401, 7.15809059112063480398815568835, 7.72109461726772599695639038795, 8.026354800464021329450976281226, 8.953040170137146208791208275335, 9.003827506150334562556987410073, 9.775061664743793348863091123208, 9.823494486222086203985184051987, 10.62477045879607834404759818228, 10.70416955903848286505051234552
