| L(s) = 1 | − 0.763i·2-s + 3.41·4-s − 2.23i·5-s + 11.6·7-s − 5.66i·8-s − 1.70·10-s − 3.31i·11-s + 7.67·13-s − 8.90i·14-s + 9.34·16-s + 6.34i·17-s − 22.3·19-s − 7.64i·20-s − 2.53·22-s + 14.5i·23-s + ⋯ |
| L(s) = 1 | − 0.381i·2-s + 0.854·4-s − 0.447i·5-s + 1.66·7-s − 0.707i·8-s − 0.170·10-s − 0.301i·11-s + 0.590·13-s − 0.635i·14-s + 0.584·16-s + 0.373i·17-s − 1.17·19-s − 0.382i·20-s − 0.115·22-s + 0.632i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.637447104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.637447104\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 11 | \( 1 + 3.31iT \) |
| good | 2 | \( 1 + 0.763iT - 4T^{2} \) |
| 7 | \( 1 - 11.6T + 49T^{2} \) |
| 13 | \( 1 - 7.67T + 169T^{2} \) |
| 17 | \( 1 - 6.34iT - 289T^{2} \) |
| 19 | \( 1 + 22.3T + 361T^{2} \) |
| 23 | \( 1 - 14.5iT - 529T^{2} \) |
| 29 | \( 1 + 2.65iT - 841T^{2} \) |
| 31 | \( 1 - 41.9T + 961T^{2} \) |
| 37 | \( 1 + 54.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 16.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 5.87T + 1.84e3T^{2} \) |
| 47 | \( 1 + 0.693iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 39.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 103. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 54.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 12.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 112. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 56.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 1.83T + 6.24e3T^{2} \) |
| 83 | \( 1 + 85.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 69.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 73.9T + 9.40e3T^{2} \) |
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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
−10.86024463745528648921202357946, −9.960716171923320136868981736872, −8.501914362986680413589097535706, −8.160467361443333345781167640175, −6.95236181034905124960055330965, −5.89999200602954778359899481745, −4.82437622428635340118495816477, −3.70796686422167518218562158042, −2.14952011548871853340207627574, −1.24366471772909060263452849285,
1.58209092753423074675261597511, 2.61884794119964663608257886574, 4.25952262585635330651040873343, 5.30788475151158347793475726431, 6.38054095249847049976458352290, 7.20837783135912088387503287447, 8.110938724405356395798576775747, 8.718825489144620962255283714837, 10.39491041102182341978645040653, 10.81315178218495856643016938448
