| L(s) = 1 | + 3.60·2-s − 50.9·4-s + 83.0i·5-s − 414.·8-s + 299. i·10-s + 442.·11-s − 696. i·13-s + 1.76e3·16-s + 6.28e3i·17-s + 2.68e3i·19-s − 4.23e3i·20-s + 1.59e3·22-s − 1.57e4·23-s + 8.72e3·25-s − 2.51e3i·26-s + ⋯ |
| L(s) = 1 | + 0.450·2-s − 0.796·4-s + 0.664i·5-s − 0.810·8-s + 0.299i·10-s + 0.332·11-s − 0.317i·13-s + 0.431·16-s + 1.28i·17-s + 0.391i·19-s − 0.529i·20-s + 0.149·22-s − 1.29·23-s + 0.558·25-s − 0.142i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.07123610806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07123610806\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3.60T + 64T^{2} \) |
| 5 | \( 1 - 83.0iT - 1.56e4T^{2} \) |
| 11 | \( 1 - 442.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 696. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.28e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 2.68e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.57e4T + 1.48e8T^{2} \) |
| 29 | \( 1 - 2.32e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 4.75e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.01e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.81e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.51e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 5.02e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.99e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.87e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.17e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 3.84e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 1.56e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 3.75e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 3.31e4T + 2.43e11T^{2} \) |
| 83 | \( 1 + 9.84e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 2.62e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 5.75e5iT - 8.32e11T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.42543911883421290561464172266, −10.04660378164772475458128761464, −8.714044253034684408863927472487, −8.161476822196522918071948049551, −6.77131856812594972325332637890, −5.98668612225982794301082146634, −4.92019562699356616486202924738, −3.85653135224836531059358550750, −3.07236362080416272995893717224, −1.52388816523098548411892061100,
0.01556310836664396114042772613, 1.02393295537140872069217436584, 2.61609326933337475897445956593, 3.93664953581520638330064637148, 4.69500354320313399435592650111, 5.53166319279733175574772706213, 6.62625490782862488833682305970, 7.87708239926876590184822185700, 8.829882343577799854262063428372, 9.409895859952079922226379305969
