| L(s) = 1 | + 4·3-s + 3·4-s + 8·9-s − 2·11-s + 12·12-s + 4·13-s + 5·16-s − 8·17-s − 2·23-s + 12·27-s + 12·29-s + 2·31-s − 8·33-s + 24·36-s − 4·37-s + 16·39-s + 2·41-s − 6·44-s + 4·47-s + 20·48-s − 32·51-s + 12·52-s − 12·61-s + 3·64-s − 4·67-s − 24·68-s − 8·69-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3/2·4-s + 8/3·9-s − 0.603·11-s + 3.46·12-s + 1.10·13-s + 5/4·16-s − 1.94·17-s − 0.417·23-s + 2.30·27-s + 2.22·29-s + 0.359·31-s − 1.39·33-s + 4·36-s − 0.657·37-s + 2.56·39-s + 0.312·41-s − 0.904·44-s + 0.583·47-s + 2.88·48-s − 4.48·51-s + 1.66·52-s − 1.53·61-s + 3/8·64-s − 0.488·67-s − 2.91·68-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 534361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 534361 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.551844985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.551844985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| 43 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.68967811705408164721969778838, −10.16783068376124761506912308083, −9.660315414897447720112299911427, −9.157415864536252235318450975672, −8.757951661733755559047528531306, −8.360892683677072043317692309069, −8.232860858247973381062938477576, −7.67574378349246143757087391030, −7.23887016971579815391748163711, −6.65366364730885702834720601995, −6.47470835122132536280224377347, −5.98565275205293651201228046299, −5.10361443357043953192677230414, −4.42070198147615999439370557529, −4.02391036294166465288780488830, −3.11478129124025493411316977348, −3.07378971741742217517660556081, −2.30334397759167169838240475932, −2.18391306438087379011073993523, −1.28023845653177343995193215651,
1.28023845653177343995193215651, 2.18391306438087379011073993523, 2.30334397759167169838240475932, 3.07378971741742217517660556081, 3.11478129124025493411316977348, 4.02391036294166465288780488830, 4.42070198147615999439370557529, 5.10361443357043953192677230414, 5.98565275205293651201228046299, 6.47470835122132536280224377347, 6.65366364730885702834720601995, 7.23887016971579815391748163711, 7.67574378349246143757087391030, 8.232860858247973381062938477576, 8.360892683677072043317692309069, 8.757951661733755559047528531306, 9.157415864536252235318450975672, 9.660315414897447720112299911427, 10.16783068376124761506912308083, 10.68967811705408164721969778838
