| L(s) = 1 | + (0.348 + 1.37i)2-s + (0.545 + 1.64i)3-s + (−1.75 + 0.955i)4-s + (−2.82 + 0.856i)5-s + (−2.06 + 1.32i)6-s + (−0.0745 + 0.374i)7-s + (−1.92 − 2.07i)8-s + (−2.40 + 1.79i)9-s + (−2.15 − 3.57i)10-s + (0.231 − 2.34i)11-s + (−2.52 − 2.36i)12-s + (3.12 + 0.949i)13-s + (−0.539 + 0.0285i)14-s + (−2.94 − 4.17i)15-s + (2.17 − 3.35i)16-s + (−0.745 + 0.308i)17-s + ⋯ |
| L(s) = 1 | + (0.246 + 0.969i)2-s + (0.314 + 0.949i)3-s + (−0.878 + 0.477i)4-s + (−1.26 + 0.383i)5-s + (−0.842 + 0.539i)6-s + (−0.0281 + 0.141i)7-s + (−0.679 − 0.733i)8-s + (−0.801 + 0.597i)9-s + (−0.682 − 1.12i)10-s + (0.0696 − 0.707i)11-s + (−0.730 − 0.683i)12-s + (0.868 + 0.263i)13-s + (−0.144 + 0.00761i)14-s + (−0.761 − 1.07i)15-s + (0.543 − 0.839i)16-s + (−0.180 + 0.0748i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.307494 - 0.794539i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.307494 - 0.794539i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.348 - 1.37i)T \) |
| 3 | \( 1 + (-0.545 - 1.64i)T \) |
| good | 5 | \( 1 + (2.82 - 0.856i)T + (4.15 - 2.77i)T^{2} \) |
| 7 | \( 1 + (0.0745 - 0.374i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.231 + 2.34i)T + (-10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-3.12 - 0.949i)T + (10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (0.745 - 0.308i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.936 + 0.500i)T + (10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (5.22 - 7.82i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.407 - 4.14i)T + (-28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (2.67 - 2.67i)T - 31iT^{2} \) |
| 37 | \( 1 + (5.34 + 2.85i)T + (20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (1.74 - 2.61i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-5.57 - 6.78i)T + (-8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (10.9 - 4.55i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-0.956 + 9.71i)T + (-51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (-11.2 + 3.40i)T + (49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (-1.86 + 2.27i)T + (-11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (-1.44 - 1.18i)T + (13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (0.884 - 4.44i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-3.73 + 0.742i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (0.353 - 0.853i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (6.40 - 3.42i)T + (46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (-5.43 + 3.63i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (8.45 + 8.45i)T + 97iT^{2} \) |
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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
−11.64546657245668574483996452457, −11.16283867057744055263309074720, −9.881022437448266711636656520908, −8.854434272623988379017255899893, −8.217255402054815433480576262915, −7.36311578604107437213163757205, −6.11462438846017894837050147383, −5.06996273840154311226803996122, −3.78798332537938199701108804219, −3.43654025546543611698127474197,
0.51148537584534636442848842165, 2.12395962549744058313011746122, 3.56953860649744176145599534873, 4.39451083164968519161623454425, 5.86679036109772507653363247464, 7.14237291613049428896512526396, 8.253593360207085538057463185683, 8.688201018567975246152294980229, 10.01735890792179961310831106157, 11.05673389974507382731119838012
