| L(s) = 1 | + 27i·3-s − 522. i·5-s − 1.30e3·7-s − 729·9-s − 2.79e3i·11-s + 7.35e3i·13-s + 1.41e4·15-s + 5.64e3·17-s + 2.02e4i·19-s − 3.53e4i·21-s − 7.95e4·23-s − 1.94e5·25-s − 1.96e4i·27-s − 1.40e5i·29-s − 3.21e5·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.86i·5-s − 1.44·7-s − 0.333·9-s − 0.633i·11-s + 0.928i·13-s + 1.07·15-s + 0.278·17-s + 0.678i·19-s − 0.832i·21-s − 1.36·23-s − 2.49·25-s − 0.192i·27-s − 1.06i·29-s − 1.93·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6550730265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6550730265\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27iT \) |
| good | 5 | \( 1 + 522. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 1.30e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.79e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 7.35e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 5.64e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.02e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 7.95e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.40e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 3.21e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.04e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 2.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.03e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.07e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.13e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.80e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.00e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 4.70e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 4.38e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.51e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.44e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.45e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 8.58e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.08e6T + 8.07e13T^{2} \) |
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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
−9.851468660024043061485700403375, −9.409801431026114005632272359886, −8.710083427075690221176179127055, −7.69260112141639428054669683153, −6.10165787220936197459231327719, −5.56500524528967768689197973481, −4.23246477479812646457795104815, −3.67559682283970249798378858756, −1.99000556502441482956933809263, −0.60294343758254603802891179520,
0.21317241365070083733338121307, 2.06299438034111372290099995320, 3.01703858368843878614067710983, 3.63104084246193768868545770378, 5.60251614432830317485439497354, 6.51175328267315973467545980630, 7.04874743090844353604453217322, 7.83652176109825614459261728014, 9.343769250875765161865881610542, 10.19939906860312323350836206328
