L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.362 + 1.69i)3-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + (1.28 − 1.16i)6-s − 0.535·7-s + 0.999·8-s + (−2.73 + 1.22i)9-s + (−0.866 − 0.499i)10-s + 5.20i·11-s + (−1.64 − 0.532i)12-s + (1.58 + 0.917i)13-s + (0.267 + 0.463i)14-s + (1.16 + 1.28i)15-s + (−0.5 − 0.866i)16-s + (−3.93 + 2.27i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.209 + 0.977i)3-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (0.524 − 0.473i)6-s − 0.202·7-s + 0.353·8-s + (−0.912 + 0.409i)9-s + (−0.273 − 0.158i)10-s + 1.57i·11-s + (−0.475 − 0.153i)12-s + (0.440 + 0.254i)13-s + (0.0715 + 0.123i)14-s + (0.299 + 0.331i)15-s + (−0.125 − 0.216i)16-s + (−0.954 + 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.818593 + 0.727363i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.818593 + 0.727363i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.362 - 1.69i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (1.25 + 4.17i)T \) |
good | 7 | \( 1 + 0.535T + 7T^{2} \) |
| 11 | \( 1 - 5.20iT - 11T^{2} \) |
| 13 | \( 1 + (-1.58 - 0.917i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.93 - 2.27i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-5.55 - 3.20i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.19 - 7.27i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.17iT - 31T^{2} \) |
| 37 | \( 1 - 5.32iT - 37T^{2} \) |
| 41 | \( 1 + (-2.33 - 4.05i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.21 - 7.30i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.52 - 2.03i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.26 - 5.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.47 + 6.01i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.11 + 7.12i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.445 + 0.257i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.29 - 5.71i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.05 - 5.29i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.65 + 4.42i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.2iT - 83T^{2} \) |
| 89 | \( 1 + (-8.15 + 14.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.567 + 0.327i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.95363494736230880403274257816, −9.917636956266590548269059706213, −9.321614662314103776766632559701, −8.785475221488960814082389660565, −7.55188297972631549165609803524, −6.41368160237058040955602968789, −4.95693300398060355921395344125, −4.35354365659327501805626216358, −3.07677902911830175415484317665, −1.86488210476021738169984578033,
0.68188495962590859525841706562, 2.33898371984952824428502066356, 3.63167121492701401928906649675, 5.45241697899379763907138014382, 6.15859423252809309171438127118, 6.86865444783784473102983106620, 7.88501937126036552861264458791, 8.688026041715512632381479882253, 9.260096800131476031335723968912, 10.66623672554499297642376141515