L(s) = 1 | + 1.40i·2-s − 2.91i·3-s + 2.02·4-s + 4.10·6-s + 11.7·7-s + 8.46i·8-s + 0.493·9-s − 6.36·11-s − 5.89i·12-s + 17.7i·13-s + 16.4i·14-s − 3.82·16-s + 17.5·17-s + 0.694i·18-s + (−16.8 + 8.86i)19-s + ⋯ |
L(s) = 1 | + 0.703i·2-s − 0.972i·3-s + 0.505·4-s + 0.683·6-s + 1.67·7-s + 1.05i·8-s + 0.0548·9-s − 0.578·11-s − 0.491i·12-s + 1.36i·13-s + 1.17i·14-s − 0.238·16-s + 1.03·17-s + 0.0385i·18-s + (−0.884 + 0.466i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.531124113\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.531124113\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (16.8 - 8.86i)T \) |
good | 2 | \( 1 - 1.40iT - 4T^{2} \) |
| 3 | \( 1 + 2.91iT - 9T^{2} \) |
| 7 | \( 1 - 11.7T + 49T^{2} \) |
| 11 | \( 1 + 6.36T + 121T^{2} \) |
| 13 | \( 1 - 17.7iT - 169T^{2} \) |
| 17 | \( 1 - 17.5T + 289T^{2} \) |
| 23 | \( 1 + 1.53T + 529T^{2} \) |
| 29 | \( 1 + 39.9iT - 841T^{2} \) |
| 31 | \( 1 - 9.24iT - 961T^{2} \) |
| 37 | \( 1 - 56.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 18.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 61.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 66.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 9.42iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 36.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 2.55iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 53.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 115.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 95.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 143.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 19.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 163. iT - 9.40e3T^{2} \) |
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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
−11.10369210620728583985458403337, −10.06842534673059749915215686914, −8.460706622664666108258165857703, −7.990296698601239941384804252353, −7.26229572966342682663620658633, −6.43090005530770517829861197569, −5.40988481117169994172956810307, −4.34385701649595412792385560137, −2.24575065261220222288493668385, −1.52732556949284498795146647539,
1.23073414611091768667503507799, 2.60053642185780321708299708485, 3.78608534525998527735241906575, 4.85719898977099108614135511836, 5.66856988449904758352505937649, 7.32293683969830873890723355585, 8.009199087674056788546993623363, 9.144152286550623173507934973388, 10.38123000363187899910801027400, 10.59820959352926309243186520325