| L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.367 + 2.08i)5-s + (−0.173 − 0.984i)6-s + (−0.5 − 2.59i)7-s + (0.500 − 0.866i)8-s + (0.173 + 0.984i)9-s + (1.62 − 1.36i)10-s + (−0.0542 − 0.0940i)11-s + (−0.500 + 0.866i)12-s + (0.542 + 3.07i)13-s + (−1.28 + 2.31i)14-s + (−1.62 + 1.36i)15-s + (−0.939 + 0.342i)16-s + (−0.853 + 4.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.442 + 0.371i)3-s + (0.0868 + 0.492i)4-s + (−0.164 + 0.932i)5-s + (−0.0708 − 0.402i)6-s + (−0.188 − 0.981i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.513 − 0.430i)10-s + (−0.0163 − 0.0283i)11-s + (−0.144 + 0.249i)12-s + (0.150 + 0.853i)13-s + (−0.343 + 0.617i)14-s + (−0.418 + 0.351i)15-s + (−0.234 + 0.0855i)16-s + (−0.206 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.975015 + 0.649300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.975015 + 0.649300i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
| 19 | \( 1 + (-3.85 + 2.03i)T \) |
| good | 5 | \( 1 + (0.367 - 2.08i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.0542 + 0.0940i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.542 - 3.07i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.853 - 4.83i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-6.26 - 2.28i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (3.83 + 1.39i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + 9.16T + 31T^{2} \) |
| 37 | \( 1 + (-4.62 - 8.00i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.19 - 12.4i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (6.01 + 5.05i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.68 - 9.56i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.0551 - 0.312i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.593 - 3.36i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.99 - 1.45i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-5.85 + 4.91i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.84 - 4.06i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (10.7 + 9.01i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-11.3 + 4.11i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (4.97 + 8.60i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.59 + 3.85i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-14.1 + 5.15i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.46116131639792045434395236811, −9.619764710882136044814608185148, −8.969027063475435968933324370952, −7.82629586174416936560673874522, −7.18199988594976874938831885804, −6.38449329799365653230868476413, −4.74349609358388156287593139366, −3.66362128592190842586567707799, −3.04525545256753929426195618536, −1.53453660880484668108193343237,
0.70000232347625481265482815606, 2.21095767582941558643767782329, 3.46995195357977599898794732111, 5.16676268122981394807518990755, 5.51944056345928062807365250954, 6.90023299606744768108101227103, 7.61183940422311353814919929195, 8.594597485238881341324915273402, 9.029233095205989191989499525320, 9.687806083419279742743806731020
