| L(s) = 1 | + (−0.539 − 1.74i)2-s + (1.37 + 1.05i)3-s + (−1.11 + 0.757i)4-s + (−0.891 − 0.134i)5-s + (1.11 − 2.96i)6-s + (1.52 − 2.16i)7-s + (−0.937 − 0.747i)8-s + (0.758 + 2.90i)9-s + (0.245 + 1.62i)10-s + (3.88 − 4.18i)11-s + (−2.32 − 0.137i)12-s + (1.39 + 0.319i)13-s + (−4.59 − 1.50i)14-s + (−1.07 − 1.12i)15-s + (−1.78 + 4.54i)16-s + (−0.352 + 4.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.381 − 1.23i)2-s + (0.791 + 0.611i)3-s + (−0.555 + 0.378i)4-s + (−0.398 − 0.0600i)5-s + (0.453 − 1.21i)6-s + (0.577 − 0.816i)7-s + (−0.331 − 0.264i)8-s + (0.252 + 0.967i)9-s + (0.0776 + 0.515i)10-s + (1.16 − 1.26i)11-s + (−0.670 − 0.0397i)12-s + (0.388 + 0.0885i)13-s + (−1.22 − 0.402i)14-s + (−0.278 − 0.291i)15-s + (−0.445 + 1.13i)16-s + (−0.0854 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.892782 - 0.757695i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.892782 - 0.757695i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.37 - 1.05i)T \) |
| 7 | \( 1 + (-1.52 + 2.16i)T \) |
| good | 2 | \( 1 + (0.539 + 1.74i)T + (-1.65 + 1.12i)T^{2} \) |
| 5 | \( 1 + (0.891 + 0.134i)T + (4.77 + 1.47i)T^{2} \) |
| 11 | \( 1 + (-3.88 + 4.18i)T + (-0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.39 - 0.319i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (0.352 - 4.70i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (5.39 - 3.11i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.46 - 0.259i)T + (22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (-2.44 - 5.08i)T + (-18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-5.42 - 3.13i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.28 + 2.91i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (0.873 - 1.09i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-2.24 - 2.81i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-1.63 + 0.503i)T + (38.8 - 26.4i)T^{2} \) |
| 53 | \( 1 + (3.89 + 5.71i)T + (-19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (0.650 - 0.0979i)T + (56.3 - 17.3i)T^{2} \) |
| 61 | \( 1 + (-5.22 + 7.66i)T + (-22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (1.45 - 2.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.43 + 5.06i)T + (-44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (2.10 - 6.81i)T + (-60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (-1.11 - 1.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.783 + 3.43i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (4.65 - 4.31i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + 9.24iT - 97T^{2} \) |
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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
−12.64708594164607414666121869942, −11.48249431285656541824831810481, −10.75074965270604884358060410401, −10.05564344134420675403098529263, −8.697581827093689006211976318671, −8.241396266905822480436020817477, −6.35740151155844815127964675552, −4.11554517405305352121914019315, −3.56333099755933921370627481644, −1.64247963193033759035006271229,
2.33456021985610953995449346374, 4.41386076351379552495456552231, 6.16632779392409549617097606496, 7.04314423603278752825491846019, 7.965635460672191326947043524328, 8.796422246256619049548641270208, 9.589810274964298920868501155359, 11.70377517649239539428516207253, 12.14178406216314835671144315036, 13.65876442570268924536550284707
