L(s) = 1 | − 1.84i·2-s + (0.901 + 2.86i)3-s + 0.582·4-s + 2.44·5-s + (5.28 − 1.66i)6-s − 1.19i·7-s − 8.47i·8-s + (−7.37 + 5.16i)9-s − 4.52i·10-s − 12.8·11-s + (0.525 + 1.66i)12-s + 11.1·13-s − 2.20·14-s + (2.20 + 7.00i)15-s − 13.3·16-s + (−11.8 + 12.1i)17-s + ⋯ |
L(s) = 1 | − 0.924i·2-s + (0.300 + 0.953i)3-s + 0.145·4-s + 0.489·5-s + (0.881 − 0.277i)6-s − 0.170i·7-s − 1.05i·8-s + (−0.819 + 0.573i)9-s − 0.452i·10-s − 1.17·11-s + (0.0437 + 0.138i)12-s + 0.858·13-s − 0.157·14-s + (0.147 + 0.467i)15-s − 0.833·16-s + (−0.698 + 0.715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31991 - 0.314435i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31991 - 0.314435i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.901 - 2.86i)T \) |
| 17 | \( 1 + (11.8 - 12.1i)T \) |
good | 2 | \( 1 + 1.84iT - 4T^{2} \) |
| 5 | \( 1 - 2.44T + 25T^{2} \) |
| 7 | \( 1 + 1.19iT - 49T^{2} \) |
| 11 | \( 1 + 12.8T + 121T^{2} \) |
| 13 | \( 1 - 11.1T + 169T^{2} \) |
| 19 | \( 1 + 2.74T + 361T^{2} \) |
| 23 | \( 1 + 14.0T + 529T^{2} \) |
| 29 | \( 1 - 38.5T + 841T^{2} \) |
| 31 | \( 1 + 44.5iT - 961T^{2} \) |
| 37 | \( 1 - 34.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 58.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.16T + 1.84e3T^{2} \) |
| 47 | \( 1 - 56.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 19.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 39.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 86.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 55.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 25.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 119. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 96.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 33.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 168. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 29.5iT - 9.40e3T^{2} \) |
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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
−15.43967003774552188952977908770, −13.85619986720817604824452375483, −12.90930175869779551329303478958, −11.32326567882887392913754491742, −10.51456215249134433980857352306, −9.677324043222778591617151460622, −8.155357159746365647926848619381, −6.03345917406229548035967412572, −4.12059314360278516823453543206, −2.49518000828973433702997610719,
2.41907705094827182736284859225, 5.53854526493286376500788493863, 6.61396316506521191868808097843, 7.79843971237691411348932966413, 8.826012414400199355798880361947, 10.74733301921740835345905364363, 12.09170761179612275507671077449, 13.44578576420665778575463327611, 14.12539482192313954232902433069, 15.47598144164648741423213015853