| L(s) = 1 | + (1.20 − 1.50i)2-s + (3.21 + 4.03i)3-s + (0.951 + 4.17i)4-s + (3.04 + 1.46i)5-s + 9.95·6-s − 15.9·7-s + (21.3 + 10.2i)8-s + (0.0787 − 0.344i)9-s + (5.86 − 2.82i)10-s + (15.5 − 68.0i)11-s + (−13.7 + 17.2i)12-s + (−7.36 − 3.54i)13-s + (−19.2 + 24.0i)14-s + (3.87 + 16.9i)15-s + (10.3 − 4.97i)16-s + (−60.7 + 29.2i)17-s + ⋯ |
| L(s) = 1 | + (0.425 − 0.533i)2-s + (0.619 + 0.776i)3-s + (0.118 + 0.521i)4-s + (0.271 + 0.130i)5-s + 0.677·6-s − 0.862·7-s + (0.943 + 0.454i)8-s + (0.00291 − 0.0127i)9-s + (0.185 − 0.0893i)10-s + (0.426 − 1.86i)11-s + (−0.331 + 0.415i)12-s + (−0.157 − 0.0757i)13-s + (−0.366 + 0.460i)14-s + (0.0667 + 0.292i)15-s + (0.161 − 0.0778i)16-s + (−0.866 + 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.90456 + 0.261327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90456 + 0.261327i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-279. + 34.0i)T \) |
| good | 2 | \( 1 + (-1.20 + 1.50i)T + (-1.78 - 7.79i)T^{2} \) |
| 3 | \( 1 + (-3.21 - 4.03i)T + (-6.00 + 26.3i)T^{2} \) |
| 5 | \( 1 + (-3.04 - 1.46i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 + 15.9T + 343T^{2} \) |
| 11 | \( 1 + (-15.5 + 68.0i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (7.36 + 3.54i)T + (1.36e3 + 1.71e3i)T^{2} \) |
| 17 | \( 1 + (60.7 - 29.2i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + (-2.53 - 11.1i)T + (-6.17e3 + 2.97e3i)T^{2} \) |
| 23 | \( 1 + (31.2 - 137. i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (-104. + 130. i)T + (-5.42e3 - 2.37e4i)T^{2} \) |
| 31 | \( 1 + (17.2 - 21.6i)T + (-6.62e3 - 2.90e4i)T^{2} \) |
| 37 | \( 1 + 32.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + (140. - 175. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 47 | \( 1 + (-45.3 - 198. i)T + (-9.35e4 + 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-14.3 + 6.91i)T + (9.28e4 - 1.16e5i)T^{2} \) |
| 59 | \( 1 + (408. - 196. i)T + (1.28e5 - 1.60e5i)T^{2} \) |
| 61 | \( 1 + (12.4 + 15.5i)T + (-5.05e4 + 2.21e5i)T^{2} \) |
| 67 | \( 1 + (89.5 + 392. i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (-195. - 854. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (868. + 418. i)T + (2.42e5 + 3.04e5i)T^{2} \) |
| 79 | \( 1 - 986.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-694. - 870. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (6.91 + 8.67i)T + (-1.56e5 + 6.87e5i)T^{2} \) |
| 97 | \( 1 + (263. - 1.15e3i)T + (-8.22e5 - 3.95e5i)T^{2} \) |
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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
−15.61772556731680384910535862785, −14.04590352047023390588962090279, −13.36971735793608897691998870092, −11.96656288194741361830407346449, −10.76067956625296037954896602779, −9.448206530399249796454028648521, −8.265190476336678253711283239495, −6.24569757258568787569309058552, −3.97680509475232690290097654402, −2.99653827147203092076714094879,
2.02764003239362303747705331267, 4.70988404892367506700030695789, 6.57811236005062942638156177830, 7.31942961071496381570991352659, 9.230520403489983745521723388821, 10.36354834217608535601384374518, 12.40177302880799927215730259785, 13.26684399454320751563720964622, 14.23652410679476178437365211262, 15.22018043597677541009290514370
