| L(s) = 1 | − 8·2-s + 14·3-s + 16·4-s − 112·6-s + 128·8-s + 81·9-s − 46·11-s + 224·12-s − 1.02e3·16-s + 1.14e3·17-s − 648·18-s − 868·19-s + 368·22-s + 1.79e3·24-s − 1.25e3·25-s + 658·27-s + 2.04e3·32-s − 644·33-s − 9.18e3·34-s + 1.29e3·36-s + 6.94e3·38-s − 1.24e3·41-s − 3.50e3·43-s − 736·44-s − 1.43e4·48-s − 4.80e3·49-s + 1.00e4·50-s + ⋯ |
| L(s) = 1 | − 2·2-s + 14/9·3-s + 4-s − 3.11·6-s + 2·8-s + 9-s − 0.380·11-s + 14/9·12-s − 4·16-s + 3.97·17-s − 2·18-s − 2.40·19-s + 0.760·22-s + 28/9·24-s − 2·25-s + 0.902·27-s + 2·32-s − 0.591·33-s − 7.94·34-s + 36-s + 4.80·38-s − 0.741·41-s − 1.89·43-s − 0.380·44-s − 6.22·48-s − 2·49-s + 4·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2818543856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2818543856\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 14 T + 115 T^{2} - 14 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 46 T + p^{4} T^{2} )^{2}( 1 - 46 T - 12525 T^{2} - 46 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 574 T + 245955 T^{2} - 574 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 434 T + 58035 T^{2} + 434 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 1246 T + p^{4} T^{2} )^{2}( 1 - 1246 T - 1273245 T^{2} - 1246 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 3502 T + p^{4} T^{2} )^{2}( 1 - 3502 T + 8845203 T^{2} - 3502 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 238 T + p^{4} T^{2} )^{2}( 1 - 238 T - 12060717 T^{2} - 238 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5134 T + p^{4} T^{2} )^{2}( 1 - 5134 T + 6206835 T^{2} - 5134 p^{4} T^{3} + p^{8} T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{4}( 1 + p^{2} T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 9506 T + 61965795 T^{2} + 9506 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2}( 1 + p^{2} T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 11186 T + 77668275 T^{2} + 11186 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 5474 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 9982 T + p^{4} T^{2} )^{2}( 1 - 9982 T + 11111043 T^{2} - 9982 p^{4} T^{3} + p^{8} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05364892948215981610385522936, −9.693399612412808229708047169069, −9.635485227422535820377368208379, −9.030243831535480027677069179175, −8.923726487406258958489816134270, −8.334542554923458423539578210860, −8.207161180523451874841366126490, −8.116766694676548186500469178137, −8.106547624685539481603952609808, −7.39209671709336525306920536719, −7.26961645617041486660887792140, −7.00987799463277227291138896052, −6.27103291761577520281832530315, −5.80363886411074850871392015906, −5.65022940945208647733538085605, −4.89127520338179107123425568710, −4.62392915783376035711623098210, −4.18781910535171885655499609603, −3.56641808130936709509242194335, −3.31145389456675216663386978292, −2.83192436061249773932247940315, −1.89600058657620902435439036041, −1.70008433285611515682936190055, −1.17464568122165450526280867946, −0.18531306804979588220954390891,
0.18531306804979588220954390891, 1.17464568122165450526280867946, 1.70008433285611515682936190055, 1.89600058657620902435439036041, 2.83192436061249773932247940315, 3.31145389456675216663386978292, 3.56641808130936709509242194335, 4.18781910535171885655499609603, 4.62392915783376035711623098210, 4.89127520338179107123425568710, 5.65022940945208647733538085605, 5.80363886411074850871392015906, 6.27103291761577520281832530315, 7.00987799463277227291138896052, 7.26961645617041486660887792140, 7.39209671709336525306920536719, 8.106547624685539481603952609808, 8.116766694676548186500469178137, 8.207161180523451874841366126490, 8.334542554923458423539578210860, 8.923726487406258958489816134270, 9.030243831535480027677069179175, 9.635485227422535820377368208379, 9.693399612412808229708047169069, 10.05364892948215981610385522936
