| L(s) = 1 | + 3·2-s + 5·4-s − 33·5-s + 99·8-s − 99·10-s + 1.13e3·11-s − 925·13-s − 459·16-s − 324·17-s − 2.31e3·19-s − 165·20-s + 3.41e3·22-s − 1.59e3·23-s − 2.38e3·25-s − 2.77e3·26-s + 2.21e3·29-s − 4.29e3·31-s − 4.36e3·32-s − 972·34-s − 1.91e4·37-s − 6.93e3·38-s − 3.26e3·40-s − 1.28e4·41-s − 2.77e3·43-s + 5.68e3·44-s − 4.78e3·46-s − 2.31e4·47-s + ⋯ |
| L(s) = 1 | + 0.530·2-s + 5/32·4-s − 0.590·5-s + 0.546·8-s − 0.313·10-s + 2.83·11-s − 1.51·13-s − 0.448·16-s − 0.271·17-s − 1.46·19-s − 0.0922·20-s + 1.50·22-s − 0.629·23-s − 0.762·25-s − 0.805·26-s + 0.489·29-s − 0.802·31-s − 0.753·32-s − 0.144·34-s − 2.29·37-s − 0.778·38-s − 0.322·40-s − 1.19·41-s − 0.228·43-s + 0.442·44-s − 0.333·46-s − 1.52·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + p^{2} T^{2} - 3 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 33 T + 3472 T^{2} + 33 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 1137 T + 645232 T^{2} - 1137 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 925 T + 951450 T^{2} + 925 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 324 T + 1502434 T^{2} + 324 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2311 T + 6241998 T^{2} + 2311 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1596 T + 7604206 T^{2} + 1596 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2217 T + 42125014 T^{2} - 2217 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4294 T + 25151367 T^{2} + 4294 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 19109 T + 184470078 T^{2} + 19109 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12858 T + 228491122 T^{2} + 12858 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2771 T + 36114396 T^{2} + 2771 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 23160 T + 570076618 T^{2} + 23160 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 31653 T + 896010526 T^{2} - 31653 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 41097 T + 896994610 T^{2} - 41097 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 42052 T + 1728407262 T^{2} + 42052 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 30763 T + 1698033324 T^{2} - 30763 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 102096 T + 6091648810 T^{2} + 102096 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 28577 T + 2636499108 T^{2} - 28577 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18464 T + 3332046261 T^{2} + 18464 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 61179 T + 8589312784 T^{2} - 61179 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 29322 T + 11382011794 T^{2} - 29322 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9791 T + 17134261944 T^{2} - 9791 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.13199842576144227096604135010, −9.712000892620380072860334888702, −9.012427557239995462750680318216, −8.823384873458009625849919547148, −8.399402264299868026734800389322, −7.66642919787712372484650821597, −7.07295666054133509186864911561, −6.87315311111943928906750493039, −6.47697299208237884606306989848, −5.92027529356180219567306530294, −5.03968112962672072890145470059, −4.75846786485297613865236781333, −3.99043836664774357386062073019, −3.95572636384637669361891774875, −3.36316645746080433214420206484, −2.24669176330310343385865794890, −1.87127008400953892077469303103, −1.31064231601552648464275750368, 0, 0,
1.31064231601552648464275750368, 1.87127008400953892077469303103, 2.24669176330310343385865794890, 3.36316645746080433214420206484, 3.95572636384637669361891774875, 3.99043836664774357386062073019, 4.75846786485297613865236781333, 5.03968112962672072890145470059, 5.92027529356180219567306530294, 6.47697299208237884606306989848, 6.87315311111943928906750493039, 7.07295666054133509186864911561, 7.66642919787712372484650821597, 8.399402264299868026734800389322, 8.823384873458009625849919547148, 9.012427557239995462750680318216, 9.712000892620380072860334888702, 10.13199842576144227096604135010
