| L(s) = 1 | + (1.17 + 0.984i)3-s + (1.76 − 0.642i)5-s + (1.03 + 1.78i)7-s + (−0.113 − 0.642i)9-s + (−1.03 + 1.78i)11-s + (1.17 − 0.984i)13-s + (2.70 + 0.984i)15-s + (0.0603 − 0.342i)17-s + (−3.75 − 2.20i)19-s + (−0.549 + 3.11i)21-s + (2.76 + 1.00i)23-s + (−1.12 + 0.943i)25-s + (2.79 − 4.84i)27-s + (1.81 + 10.3i)29-s + (−2.72 − 4.72i)31-s + ⋯ |
| L(s) = 1 | + (0.677 + 0.568i)3-s + (0.789 − 0.287i)5-s + (0.390 + 0.675i)7-s + (−0.0377 − 0.214i)9-s + (−0.311 + 0.538i)11-s + (0.325 − 0.273i)13-s + (0.698 + 0.254i)15-s + (0.0146 − 0.0829i)17-s + (−0.862 − 0.506i)19-s + (−0.119 + 0.679i)21-s + (0.576 + 0.209i)23-s + (−0.224 + 0.188i)25-s + (0.538 − 0.932i)27-s + (0.337 + 1.91i)29-s + (−0.489 − 0.848i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74156 + 0.445094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74156 + 0.445094i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (3.75 + 2.20i)T \) |
| good | 3 | \( 1 + (-1.17 - 0.984i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-1.76 + 0.642i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.03 - 1.78i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.03 - 1.78i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.17 + 0.984i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0603 + 0.342i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.76 - 1.00i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.81 - 10.3i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.72 + 4.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.51T + 37T^{2} \) |
| 41 | \( 1 + (7.64 + 6.41i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (6.52 - 2.37i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.897 + 5.09i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (7.29 + 2.65i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.0457 + 0.259i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-9.97 - 3.63i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.22 + 6.93i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (6.07 - 2.20i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (11.7 + 9.88i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (9.64 + 8.09i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.54 - 11.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.52 - 2.95i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.301 + 1.71i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.80922735174634483438263052134, −10.67747393934186889655095343555, −9.750610129556652121247281756051, −8.985672397640899094192452201933, −8.364236251248174118536038699758, −6.91551279823182885945000401928, −5.66609997652178681822075172657, −4.73049670422462457205255306697, −3.28403664517903785828647713578, −1.98063053887870594816886670629,
1.66699722584734515254755754282, 2.87397163399326601153781866814, 4.41347482171495054831622694887, 5.84988780397646688405440581318, 6.82537473069416461545284177355, 7.955443098842141423915484039862, 8.550935805518023557288474092815, 9.859462207339158524896176970248, 10.64577790784327297364220695088, 11.53391572085981320123621813119
