L(s) = 1 | + 2.36i·2-s − 3.55i·3-s − 1.59·4-s + 8.41·6-s − 11.0·7-s + 5.68i·8-s − 3.64·9-s + 16.7·11-s + 5.67i·12-s − 14.3i·13-s − 26.2i·14-s − 19.8·16-s + 20.2·17-s − 8.61i·18-s + (−16.4 − 9.51i)19-s + ⋯ |
L(s) = 1 | + 1.18i·2-s − 1.18i·3-s − 0.399·4-s + 1.40·6-s − 1.58·7-s + 0.710i·8-s − 0.404·9-s + 1.52·11-s + 0.472i·12-s − 1.10i·13-s − 1.87i·14-s − 1.23·16-s + 1.18·17-s − 0.478i·18-s + (−0.865 − 0.500i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.526389086\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526389086\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (16.4 + 9.51i)T \) |
good | 2 | \( 1 - 2.36iT - 4T^{2} \) |
| 3 | \( 1 + 3.55iT - 9T^{2} \) |
| 7 | \( 1 + 11.0T + 49T^{2} \) |
| 11 | \( 1 - 16.7T + 121T^{2} \) |
| 13 | \( 1 + 14.3iT - 169T^{2} \) |
| 17 | \( 1 - 20.2T + 289T^{2} \) |
| 23 | \( 1 - 13.5T + 529T^{2} \) |
| 29 | \( 1 + 30.3iT - 841T^{2} \) |
| 31 | \( 1 + 16.1iT - 961T^{2} \) |
| 37 | \( 1 + 51.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 74.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 6.23T + 1.84e3T^{2} \) |
| 47 | \( 1 - 44.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 30.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 17.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 20.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 7.48iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 29.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 8.78iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 63.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 162. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 75.5iT - 9.40e3T^{2} \) |
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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
−10.68127940979448827587876920287, −9.519068140990672644488372981217, −8.684376474981555082976825121722, −7.55480703705317134643084771864, −7.03495884908785397307755283337, −6.22812646600217082474746739212, −5.73128298500213874278397949594, −3.87481146581601675584597158302, −2.47175550392047726328496129001, −0.67201494699264613970854563488,
1.40760963140939299703339722853, 3.19889084983765360749122782280, 3.66909026815544875517417204865, 4.61421178301317730203674562429, 6.32347291094455697687068877119, 6.87805639068679004354365406328, 8.853603793489960740004009032637, 9.549717271353841523012332278546, 9.913853741140368534270363753025, 10.74417178295774545564921316140