| L(s) = 1 | − 2.75i·2-s − 12.3i·3-s + 8.40·4-s − 21.9i·5-s − 33.9·6-s + 63.3i·7-s − 67.2i·8-s − 71.0·9-s − 60.5·10-s − 1.17·11-s − 103. i·12-s − 173.·13-s + 174.·14-s − 271.·15-s − 51.0·16-s + 469.·17-s + ⋯ |
| L(s) = 1 | − 0.689i·2-s − 1.37i·3-s + 0.525·4-s − 0.879i·5-s − 0.944·6-s + 1.29i·7-s − 1.05i·8-s − 0.877·9-s − 0.605·10-s − 0.00967·11-s − 0.719i·12-s − 1.02·13-s + 0.890·14-s − 1.20·15-s − 0.199·16-s + 1.62·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.557954 - 1.57290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.557954 - 1.57290i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.43e3 - 1.16e3i)T \) |
| good | 2 | \( 1 + 2.75iT - 16T^{2} \) |
| 3 | \( 1 + 12.3iT - 81T^{2} \) |
| 5 | \( 1 + 21.9iT - 625T^{2} \) |
| 7 | \( 1 - 63.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 1.17T + 1.46e4T^{2} \) |
| 13 | \( 1 + 173.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 469.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 27.8iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 79.3T + 2.79e5T^{2} \) |
| 29 | \( 1 - 696. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.19e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.53e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 2.45e3T + 2.82e6T^{2} \) |
| 47 | \( 1 - 1.69e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 1.99e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.56e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 5.44e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 4.93e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 9.66e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.60e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 1.14e4T + 3.89e7T^{2} \) |
| 83 | \( 1 - 5.52e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 2.55e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.65e3T + 8.85e7T^{2} \) |
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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
−14.60894644641096190324918832258, −12.94442810054998090173961859859, −12.27838502056889228197313882025, −11.82387953545417243493547449506, −9.880935316537476201243816783775, −8.367753879461688962762470226169, −7.04123679637500052230556178248, −5.51993191471330613894584971890, −2.62780511990073890195166163342, −1.22828245147751661481197054843,
3.26704972758665571293883281250, 4.96499534700002864458523298426, 6.73123257422785286314077517651, 7.84037387907704472559513336440, 9.994771621373234759823917643754, 10.47636010720746538350222720022, 11.74919971998207822593983085432, 13.98904015320510274380246628740, 14.78979826804969102890607750265, 15.55341630590062958172333336082
