| L(s) = 1 | − 3·2-s − 2·4-s − 24·7-s + 15·8-s − 24·11-s + 91·13-s + 72·14-s − 15·16-s + 117·17-s − 198·19-s + 72·22-s − 78·23-s + 247·25-s − 273·26-s + 48·28-s − 141·29-s + 120·32-s − 351·34-s − 249·37-s + 594·38-s − 471·41-s − 104·43-s + 48·44-s + 234·46-s + 41·49-s − 741·50-s − 182·52-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 1/4·4-s − 1.29·7-s + 0.662·8-s − 0.657·11-s + 1.94·13-s + 1.37·14-s − 0.234·16-s + 1.66·17-s − 2.39·19-s + 0.697·22-s − 0.707·23-s + 1.97·25-s − 2.05·26-s + 0.323·28-s − 0.902·29-s + 0.662·32-s − 1.77·34-s − 1.10·37-s + 2.53·38-s − 1.79·41-s − 0.368·43-s + 0.164·44-s + 0.750·46-s + 0.119·49-s − 2.09·50-s − 0.485·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4112340173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4112340173\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 7 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 11 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 247 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 24 T + 535 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 24 T + 1523 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 117 T + 8776 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 198 T + 19927 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 78 T - 6083 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 141 T - 4508 T^{2} + 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 249 T + 71320 T^{2} + 249 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 471 T + 142868 T^{2} + 471 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 104 T - 68691 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 116818 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 93 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 492 T + 286067 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 1362 T + 919111 T^{2} + 1362 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 1830 T + 1474211 T^{2} + 1830 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 567359 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1276 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 519766 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1692 T + 1659257 T^{2} - 1692 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 348 T + 953041 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.20717528902327138288559179387, −13.03924393388091608205364278754, −12.22521295036010144953983761294, −11.88863427732678576769302060962, −10.76225884822375867593847032029, −10.49570431799921544407060974096, −10.30991335696103781081526290736, −9.486748251421329213926989668180, −8.991786613071356610945517737969, −8.516998791206060649721819729527, −8.267858943814520333303857331635, −7.45343929781162028545063759085, −6.43722383010280546816219765876, −6.35833433875422507205743815823, −5.46227900219245376534047894049, −4.55750188168561727517801399791, −3.59766407506016470266247373318, −3.13455995609953690567489845437, −1.64447917202708820271422566668, −0.42098454798999874767220131192,
0.42098454798999874767220131192, 1.64447917202708820271422566668, 3.13455995609953690567489845437, 3.59766407506016470266247373318, 4.55750188168561727517801399791, 5.46227900219245376534047894049, 6.35833433875422507205743815823, 6.43722383010280546816219765876, 7.45343929781162028545063759085, 8.267858943814520333303857331635, 8.516998791206060649721819729527, 8.991786613071356610945517737969, 9.486748251421329213926989668180, 10.30991335696103781081526290736, 10.49570431799921544407060974096, 10.76225884822375867593847032029, 11.88863427732678576769302060962, 12.22521295036010144953983761294, 13.03924393388091608205364278754, 13.20717528902327138288559179387
