L(s) = 1 | − 2.23i·2-s + 2.23i·3-s − 3.00·4-s − 3·5-s + 5.00·6-s + 2·7-s + 2.23i·8-s − 2.00·9-s + 6.70i·10-s − 2.23i·11-s − 6.70i·12-s − 13-s − 4.47i·14-s − 6.70i·15-s − 0.999·16-s + 4.47i·17-s + ⋯ |
L(s) = 1 | − 1.58i·2-s + 1.29i·3-s − 1.50·4-s − 1.34·5-s + 2.04·6-s + 0.755·7-s + 0.790i·8-s − 0.666·9-s + 2.12i·10-s − 0.674i·11-s − 1.93i·12-s − 0.277·13-s − 1.19i·14-s − 1.73i·15-s − 0.249·16-s + 1.08i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.543733 - 0.289994i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.543733 - 0.289994i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (3 + 4.47i)T \) |
good | 2 | \( 1 + 2.23iT - 2T^{2} \) |
| 3 | \( 1 - 2.23iT - 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 31 | \( 1 + 6.70iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 - 6.70iT - 43T^{2} \) |
| 47 | \( 1 + 2.23iT - 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 6.70iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 4.47iT - 89T^{2} \) |
| 97 | \( 1 + 13.4iT - 97T^{2} \) |
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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
−16.99778015087597905058112764402, −15.64798762633737740755367231951, −14.71890083440623415685845527252, −12.86797184750678925981940018018, −11.36178678057320903112498398069, −11.05371410469819666640288849456, −9.643088170761776087576267011655, −8.215421025489560642376724408658, −4.60354096694864069732607307798, −3.55916987895440262001177104646,
4.92263227468655885794947561274, 7.07243239456893174212750743030, 7.45878499055687683322781099288, 8.662965289571010527502610511555, 11.47438908687119495406679323194, 12.63899564306666386775870220650, 14.07619189428567131477170846308, 15.06413860942384192522890506409, 16.06488766482125120963498063711, 17.39588301424158933875677218648