| L(s) = 1 | + (1.16 − 3.82i)2-s + (−13.2 − 8.92i)4-s + 39.0·5-s + 12.2i·7-s + (−49.6 + 40.4i)8-s + (45.5 − 149. i)10-s + 111. i·11-s + 208.·13-s + (46.6 + 14.2i)14-s + (96.6 + 237. i)16-s − 93.3·17-s + 26.8i·19-s + (−518. − 348. i)20-s + (424. + 129. i)22-s + 874. i·23-s + ⋯ |
| L(s) = 1 | + (0.291 − 0.956i)2-s + (−0.829 − 0.557i)4-s + 1.56·5-s + 0.249i·7-s + (−0.775 + 0.631i)8-s + (0.455 − 1.49i)10-s + 0.917i·11-s + 1.23·13-s + (0.238 + 0.0726i)14-s + (0.377 + 0.925i)16-s − 0.323·17-s + 0.0744i·19-s + (−1.29 − 0.871i)20-s + (0.877 + 0.267i)22-s + 1.65i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.557i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.829 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.899328317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.899328317\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.16 + 3.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 39.0T + 625T^{2} \) |
| 7 | \( 1 - 12.2iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 111. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 208.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 93.3T + 8.35e4T^{2} \) |
| 19 | \( 1 - 26.8iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 874. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.30e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 685. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.76e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 78.0T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.62e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.30e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.31e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 5.56e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.18e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 246. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.60e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.56e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.18e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.87e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.16e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 5.73e3T + 8.85e7T^{2} \) |
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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
−10.76081960061673723242046836105, −10.04219584684015950645292862286, −9.324384142233706887207037362888, −8.490406914101075668369614426620, −6.69021600770837742602337021187, −5.73025108533868679362425105061, −4.89624156797015776154541058677, −3.43922923471703314652389210188, −2.12806342895558660268055648110, −1.32383160913836763596528908516,
0.897186080992715486453342949868, 2.69993019005585562029312801632, 4.14084911315653573750851738665, 5.43440008585809505924371184486, 6.17186253018194560190373077652, 6.82946447804606150925652136994, 8.447105533523974079823091956778, 8.848695874295487726568668489933, 10.05132077691533264459492855345, 10.83548381800524200819765191604
