| L(s) = 1 | − 2.34i·2-s − 5.73i·3-s − 1.51·4-s − 13.4·6-s − 9.56·7-s − 5.83i·8-s − 23.9·9-s + 4.15·11-s + 8.68i·12-s − 4.55i·13-s + 22.4i·14-s − 19.7·16-s + 27.9·17-s + 56.1i·18-s + (7.04 − 17.6i)19-s + ⋯ |
| L(s) = 1 | − 1.17i·2-s − 1.91i·3-s − 0.378·4-s − 2.24·6-s − 1.36·7-s − 0.729i·8-s − 2.65·9-s + 0.377·11-s + 0.723i·12-s − 0.350i·13-s + 1.60i·14-s − 1.23·16-s + 1.64·17-s + 3.11i·18-s + (0.370 − 0.928i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.099444799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099444799\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + (-7.04 + 17.6i)T \) |
| good | 2 | \( 1 + 2.34iT - 4T^{2} \) |
| 3 | \( 1 + 5.73iT - 9T^{2} \) |
| 7 | \( 1 + 9.56T + 49T^{2} \) |
| 11 | \( 1 - 4.15T + 121T^{2} \) |
| 13 | \( 1 + 4.55iT - 169T^{2} \) |
| 17 | \( 1 - 27.9T + 289T^{2} \) |
| 23 | \( 1 - 13.9T + 529T^{2} \) |
| 29 | \( 1 + 14.6iT - 841T^{2} \) |
| 31 | \( 1 - 28.9iT - 961T^{2} \) |
| 37 | \( 1 + 30.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 44.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 69.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 30.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 54.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 99.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 3.05iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 23.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 21.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 77.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 93.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 126. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 77.2iT - 9.40e3T^{2} \) |
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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
−10.15751585793215482840039501040, −9.352635436003847128353781222745, −8.219087171061088696615553144431, −7.04927465274878912912366053874, −6.65730120617354490515679299705, −5.54522101310656740447355633420, −3.32636692966797834012711600333, −2.82663354375394769255600925051, −1.44242232215690719845716332269, −0.45641377103411013123135731274,
3.06160665489974597845702390618, 3.84145022836352269436104085205, 5.14723621357451024162140037326, 5.80439954188646801488226010558, 6.70488952700975395181783245552, 7.995867253428655962527752003922, 8.938944132711700304061679569271, 9.735049365600413868089973858680, 10.21140380862684824373153416523, 11.35347072412283913263861052510
