L(s) = 1 | − 3.76i·2-s − 4.52i·3-s − 10.1·4-s + 3.85i·5-s − 17.0·6-s − 7.12i·7-s + 23.1i·8-s − 11.5·9-s + 14.4·10-s − 3.33·11-s + 46.0i·12-s − 18.1·13-s − 26.8·14-s + 17.4·15-s + 46.5·16-s + 27.0i·17-s + ⋯ |
L(s) = 1 | − 1.88i·2-s − 1.50i·3-s − 2.54·4-s + 0.770i·5-s − 2.84·6-s − 1.01i·7-s + 2.89i·8-s − 1.27·9-s + 1.44·10-s − 0.303·11-s + 3.83i·12-s − 1.39·13-s − 1.91·14-s + 1.16·15-s + 2.91·16-s + 1.59i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1691818100\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1691818100\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 + (532. + 126. i)T \) |
good | 2 | \( 1 + 3.76iT - 4T^{2} \) |
| 3 | \( 1 + 4.52iT - 9T^{2} \) |
| 5 | \( 1 - 3.85iT - 25T^{2} \) |
| 7 | \( 1 + 7.12iT - 49T^{2} \) |
| 11 | \( 1 + 3.33T + 121T^{2} \) |
| 13 | \( 1 + 18.1T + 169T^{2} \) |
| 17 | \( 1 - 27.0iT - 289T^{2} \) |
| 19 | \( 1 - 23.2T + 361T^{2} \) |
| 23 | \( 1 + 17.8iT - 529T^{2} \) |
| 29 | \( 1 + 50.9T + 841T^{2} \) |
| 31 | \( 1 + 9.91iT - 961T^{2} \) |
| 37 | \( 1 + 42.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 2.94iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 41.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 7.50T + 2.20e3T^{2} \) |
| 53 | \( 1 + 76.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 89.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 36.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 117.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 4.68iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 73.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 54.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 79.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 54.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 175.T + 9.40e3T^{2} \) |
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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
−10.13637332135061700597291252614, −9.037210398982181298654913426522, −7.72381080980757159900501535741, −7.32380567421043425019807833098, −5.91909861840153271705698488581, −4.47373508623821181383134733987, −3.28691370980161968438971625650, −2.33309874835630665122161371184, −1.35308891105555547484827261002, −0.06952287404323233222427587052,
3.22766829454177595386803844267, 4.68393232867708095765978633382, 5.15327701722088499217838454858, 5.55465519943317783712825032443, 7.10287589221705619509187808992, 7.908426815998732139297888686594, 9.085890143043168716264854108964, 9.322619598377771750618317088080, 9.924399719232545283738020033365, 11.51816756215796885444192113086