| L(s) = 1 | + (1.26 + 0.460i)2-s + (−2.70 − 0.984i)3-s + (−0.141 − 0.118i)4-s + (−0.673 + 0.565i)5-s + (−2.97 − 2.49i)6-s + (−1.47 − 2.54i)8-s + (4.05 + 3.40i)9-s + (−1.11 + 0.405i)10-s + (1.11 − 1.92i)11-s + (0.266 + 0.460i)12-s + (−1.97 − 1.65i)13-s + (2.37 − 0.866i)15-s + (−0.624 − 3.54i)16-s + (−0.358 + 0.300i)17-s + (3.56 + 6.17i)18-s + (−2.77 + 3.35i)19-s + ⋯ |
| L(s) = 1 | + (0.895 + 0.325i)2-s + (−1.56 − 0.568i)3-s + (−0.0707 − 0.0593i)4-s + (−0.301 + 0.252i)5-s + (−1.21 − 1.01i)6-s + (−0.520 − 0.901i)8-s + (1.35 + 1.13i)9-s + (−0.352 + 0.128i)10-s + (0.335 − 0.581i)11-s + (0.0768 + 0.133i)12-s + (−0.546 − 0.458i)13-s + (0.614 − 0.223i)15-s + (−0.156 − 0.885i)16-s + (−0.0869 + 0.0729i)17-s + (0.840 + 1.45i)18-s + (−0.637 + 0.770i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.285753 + 0.378189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.285753 + 0.378189i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + (2.77 - 3.35i)T \) |
| good | 2 | \( 1 + (-1.26 - 0.460i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (2.70 + 0.984i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (0.673 - 0.565i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (-1.11 + 1.92i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.97 + 1.65i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.358 - 0.300i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.467 - 2.65i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (1.19 - 6.77i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 + (2.47 - 4.28i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.89 - 1.59i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.66 + 1.33i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-5.58 - 4.68i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (2.17 + 1.82i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-4.83 + 4.05i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.58 - 8.98i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (7.21 - 2.62i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.74 - 3.18i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.30 - 0.475i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (2.05 + 11.6i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (7.41 - 12.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (9.67 - 3.51i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (1.64 + 9.30i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.51901087312716888329699472937, −9.760024276002513095764897279812, −8.462774742876983263024994781057, −7.28422462552910049990377745773, −6.64849602411039563589927777127, −5.85920680815967867224260128531, −5.30347947955184391699108033451, −4.39727684691027067249768259083, −3.28496802669240283530761374371, −1.25034219362912372278718982149,
0.22552770491951505481854021575, 2.38257447769699727631038592325, 4.03943591942922162903130136779, 4.47977574253035898088583003273, 5.12575153990328347994548933770, 6.12882866709044161860793091127, 6.84895738943231565636523348214, 8.166722423227128457468259060209, 9.199234799233811005991736799311, 10.07666465212397074357971018894
