| L(s) = 1 | − 8i·5-s − 12·7-s + 12i·11-s + 20i·13-s − 62·17-s − 108i·19-s − 72·23-s + 61·25-s + 128i·29-s + 204·31-s + 96i·35-s + 228i·37-s + 22·41-s − 204i·43-s − 600·47-s + ⋯ |
| L(s) = 1 | − 0.715i·5-s − 0.647·7-s + 0.328i·11-s + 0.426i·13-s − 0.884·17-s − 1.30i·19-s − 0.652·23-s + 0.487·25-s + 0.819i·29-s + 1.18·31-s + 0.463i·35-s + 1.01i·37-s + 0.0838·41-s − 0.723i·43-s − 1.86·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.308962664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308962664\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8iT - 125T^{2} \) |
| 7 | \( 1 + 12T + 343T^{2} \) |
| 11 | \( 1 - 12iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 62T + 4.91e3T^{2} \) |
| 19 | \( 1 + 108iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 72T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 204T + 2.97e4T^{2} \) |
| 37 | \( 1 - 228iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 22T + 6.89e4T^{2} \) |
| 43 | \( 1 + 204iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 600T + 1.03e5T^{2} \) |
| 53 | \( 1 - 256iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 828iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 84iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 348iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 456T + 3.57e5T^{2} \) |
| 73 | \( 1 - 822T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.35e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 108iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 938T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.27e3T + 9.12e5T^{2} \) |
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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
−9.372308067208420583702570582174, −8.857629011517218420482599704623, −7.995401160258820462855624880220, −6.82003486529962340353003755327, −6.39728605477763677591452819786, −5.01690214404273040329892142925, −4.52410848442112373346624290907, −3.28264206524264046864205554789, −2.16873219993995176586819503290, −0.840451307738406061285524141273,
0.40144566781329716112066346806, 2.02358474661958223288999036246, 3.08850997195347398361109585304, 3.86877645535031687897918520987, 5.06765661646327287209967904845, 6.31103761353169741421812367486, 6.49449998065306935175984411500, 7.78577175595520483713130097474, 8.343062224615406398777490456370, 9.537632404995594383477327956415
