| L(s) = 1 | + 2·2-s − 10·4-s − 2·5-s − 20·7-s − 32·8-s − 4·10-s − 80·13-s − 40·14-s + 44·16-s − 124·17-s − 72·19-s + 20·20-s + 98·23-s − 55·25-s − 160·26-s + 200·28-s + 144·29-s − 34·31-s + 248·32-s − 248·34-s + 40·35-s + 54·37-s − 144·38-s + 64·40-s + 536·41-s + 60·43-s + 196·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 5/4·4-s − 0.178·5-s − 1.07·7-s − 1.41·8-s − 0.126·10-s − 1.70·13-s − 0.763·14-s + 0.687·16-s − 1.76·17-s − 0.869·19-s + 0.223·20-s + 0.888·23-s − 0.439·25-s − 1.20·26-s + 1.34·28-s + 0.922·29-s − 0.196·31-s + 1.37·32-s − 1.25·34-s + 0.193·35-s + 0.239·37-s − 0.614·38-s + 0.252·40-s + 2.04·41-s + 0.212·43-s + 0.628·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.365791406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.365791406\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 7 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 59 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 738 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 80 T + 4794 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 124 T + 13238 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 72 T + 4214 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 98 T + 22847 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 144 T + 44554 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 34 T + 57519 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 54 T + 101843 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 536 T + 209618 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 60 T + 159146 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 272 T + 182942 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 492 T + 348862 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 634 T + 458975 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 840 T + 528794 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 754 T + 742455 T^{2} - 754 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 678 T + 813415 T^{2} - 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 400 T + 160962 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 p T - 279966 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 468 T + 1155130 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1842 T + 1935427 T^{2} - 1842 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2194 T + 2966547 T^{2} - 2194 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.535121418063040927032227916030, −9.268029856583904660769443842851, −9.031771068047687210540433210776, −8.669646720438419843933949861541, −7.896441982577998042654763489068, −7.80613416626232424163215119239, −6.95663755294698246279681694205, −6.83179115483115673144893828805, −6.07283923150480295926549779891, −6.03699728062911336832084269172, −5.16796055393739913921415526236, −4.85052430661207449830518407440, −4.40654447738027405002660176216, −4.25312756537127109823051366171, −3.53478758078441822082881828915, −3.06404389116835686225141211835, −2.46775883914995997535553971467, −2.04916937419102280263668077919, −0.55642425167117287909855445106, −0.48450459504220144532868062632,
0.48450459504220144532868062632, 0.55642425167117287909855445106, 2.04916937419102280263668077919, 2.46775883914995997535553971467, 3.06404389116835686225141211835, 3.53478758078441822082881828915, 4.25312756537127109823051366171, 4.40654447738027405002660176216, 4.85052430661207449830518407440, 5.16796055393739913921415526236, 6.03699728062911336832084269172, 6.07283923150480295926549779891, 6.83179115483115673144893828805, 6.95663755294698246279681694205, 7.80613416626232424163215119239, 7.896441982577998042654763489068, 8.669646720438419843933949861541, 9.031771068047687210540433210776, 9.268029856583904660769443842851, 9.535121418063040927032227916030
