| L(s) = 1 | + 2·2-s + 9·5-s − 8·8-s + 18·10-s − 57·11-s + 140·13-s − 16·16-s − 51·17-s + 5·19-s − 114·22-s + 69·23-s + 125·25-s + 280·26-s − 228·29-s + 23·31-s − 102·34-s + 253·37-s + 10·38-s − 72·40-s − 84·41-s − 248·43-s + 138·46-s − 201·47-s + 250·50-s − 393·53-s − 513·55-s − 456·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.804·5-s − 0.353·8-s + 0.569·10-s − 1.56·11-s + 2.98·13-s − 1/4·16-s − 0.727·17-s + 0.0603·19-s − 1.10·22-s + 0.625·23-s + 25-s + 2.11·26-s − 1.45·29-s + 0.133·31-s − 0.514·34-s + 1.12·37-s + 0.0426·38-s − 0.284·40-s − 0.319·41-s − 0.879·43-s + 0.442·46-s − 0.623·47-s + 0.707·50-s − 1.01·53-s − 1.25·55-s − 1.03·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.495692687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.495692687\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T - 44 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 57 T + 1918 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 p T - 8 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T - 6834 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 p T - 14 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 114 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 23 T - 29262 T^{2} - 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 253 T + 13356 T^{2} - 253 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 124 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 201 T - 63422 T^{2} + 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 393 T + 5572 T^{2} + 393 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 219 T - 157418 T^{2} + 219 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 709 T + 275700 T^{2} + 709 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 419 T - 125202 T^{2} + 419 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 96 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 313 T - 291048 T^{2} + 313 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 461 T - 280518 T^{2} + 461 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 588 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1017 T + 329320 T^{2} - 1017 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 1834 T + p^{3} T^{2} )^{2} \) |
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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
−10.22248605004855437213955097624, −9.315033098008783370951175670099, −9.147409219671262156311145214390, −8.785101259799875944939989603785, −8.326203716411099886365180275606, −7.81354504870175813919754011897, −7.53449368960982168236250686623, −6.64156583006287359659093383740, −6.39115237548035909812671913911, −6.01412956470449521866673363870, −5.66057308737277947206967451655, −4.98074150014480016222205894316, −4.88932624879956249380394581299, −3.98379792382082876720006959059, −3.71313759502123318246440156023, −2.89158597003839989872205287195, −2.80326950899123060341505491075, −1.67020384035870446609960761170, −1.44475935147327942645373198978, −0.42438150281191247574806351189,
0.42438150281191247574806351189, 1.44475935147327942645373198978, 1.67020384035870446609960761170, 2.80326950899123060341505491075, 2.89158597003839989872205287195, 3.71313759502123318246440156023, 3.98379792382082876720006959059, 4.88932624879956249380394581299, 4.98074150014480016222205894316, 5.66057308737277947206967451655, 6.01412956470449521866673363870, 6.39115237548035909812671913911, 6.64156583006287359659093383740, 7.53449368960982168236250686623, 7.81354504870175813919754011897, 8.326203716411099886365180275606, 8.785101259799875944939989603785, 9.147409219671262156311145214390, 9.315033098008783370951175670099, 10.22248605004855437213955097624
