| L(s) = 1 | + 4·2-s − 10·3-s − 84·5-s − 40·6-s − 64·8-s + 243·9-s − 336·10-s + 336·11-s + 1.16e3·13-s + 840·15-s − 256·16-s + 1.45e3·17-s + 972·18-s − 470·19-s + 1.34e3·22-s + 4.20e3·23-s + 640·24-s + 3.12e3·25-s + 4.67e3·26-s − 6.29e3·27-s + 9.73e3·29-s + 3.36e3·30-s + 7.37e3·31-s − 3.36e3·33-s + 5.83e3·34-s − 1.43e4·37-s − 1.88e3·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.641·3-s − 1.50·5-s − 0.453·6-s − 0.353·8-s + 9-s − 1.06·10-s + 0.837·11-s + 1.91·13-s + 0.963·15-s − 1/4·16-s + 1.22·17-s + 0.707·18-s − 0.298·19-s + 0.592·22-s + 1.65·23-s + 0.226·24-s + 25-s + 1.35·26-s − 1.66·27-s + 2.14·29-s + 0.681·30-s + 1.37·31-s − 0.537·33-s + 0.865·34-s − 1.72·37-s − 0.211·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.890890804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.890890804\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T - 143 T^{2} + 10 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 84 T + 3931 T^{2} + 84 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 336 T - 48155 T^{2} - 336 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 584 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1458 T + 705907 T^{2} - 1458 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 470 T - 2255199 T^{2} + 470 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4200 T + 11203657 T^{2} - 4200 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4866 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 7372 T + 25717233 T^{2} - 7372 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 14330 T + 136004943 T^{2} + 14330 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6222 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 3704 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1812 T - 226061663 T^{2} - 1812 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 37242 T + 968771071 T^{2} - 37242 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 34302 T + 461702905 T^{2} + 34302 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24476 T - 245521725 T^{2} + 24476 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 17452 T - 1045552803 T^{2} - 17452 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 28224 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3602 T - 2060097189 T^{2} + 3602 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 42872 T - 1239048015 T^{2} + 42872 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 35202 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 26730 T - 4869566549 T^{2} + 26730 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 16978 T + p^{5} T^{2} )^{2} \) |
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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
−13.14259625871645937250728344062, −12.56882496479216782455908794931, −12.03905232296589243447536208231, −11.93817630139947359479010393122, −11.10586620967538715037104926787, −10.89263815807822240149240516640, −10.17690052465870558289914653280, −9.445144352133877543342587753048, −8.501014717310190225776505175163, −8.461487243225759803942694025533, −7.44199405538043889081754776753, −6.95997025989419267056742272010, −6.28277895233798925854182952573, −5.70288037548559223135520407120, −4.76501827757681356180032702853, −4.22188121943707209568416426724, −3.69738360073889846122990808383, −3.07045766800819100790285523428, −1.22807131874600902171485289198, −0.77764182036053384191011107479,
0.77764182036053384191011107479, 1.22807131874600902171485289198, 3.07045766800819100790285523428, 3.69738360073889846122990808383, 4.22188121943707209568416426724, 4.76501827757681356180032702853, 5.70288037548559223135520407120, 6.28277895233798925854182952573, 6.95997025989419267056742272010, 7.44199405538043889081754776753, 8.461487243225759803942694025533, 8.501014717310190225776505175163, 9.445144352133877543342587753048, 10.17690052465870558289914653280, 10.89263815807822240149240516640, 11.10586620967538715037104926787, 11.93817630139947359479010393122, 12.03905232296589243447536208231, 12.56882496479216782455908794931, 13.14259625871645937250728344062
