L(s) = 1 | + (0.408 + 0.235i)2-s + (−0.888 − 1.53i)4-s + (2.19 + 0.441i)5-s − 1.17i·7-s − 1.78i·8-s + (0.791 + 0.697i)10-s − 0.713·11-s + (3.55 − 2.05i)13-s + (0.277 − 0.480i)14-s + (−1.35 + 2.34i)16-s + (−2.21 − 1.27i)17-s + (−1.57 − 4.06i)19-s + (−1.26 − 3.76i)20-s + (−0.291 − 0.168i)22-s + (0.525 − 0.303i)23-s + ⋯ |
L(s) = 1 | + (0.288 + 0.166i)2-s + (−0.444 − 0.769i)4-s + (0.980 + 0.197i)5-s − 0.444i·7-s − 0.630i·8-s + (0.250 + 0.220i)10-s − 0.215·11-s + (0.985 − 0.569i)13-s + (0.0741 − 0.128i)14-s + (−0.339 + 0.587i)16-s + (−0.536 − 0.309i)17-s + (−0.360 − 0.932i)19-s + (−0.283 − 0.842i)20-s + (−0.0621 − 0.0358i)22-s + (0.109 − 0.0632i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.51484 - 1.02331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.51484 - 1.02331i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.19 - 0.441i)T \) |
| 19 | \( 1 + (1.57 + 4.06i)T \) |
good | 2 | \( 1 + (-0.408 - 0.235i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + 1.17iT - 7T^{2} \) |
| 11 | \( 1 + 0.713T + 11T^{2} \) |
| 13 | \( 1 + (-3.55 + 2.05i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.21 + 1.27i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.525 + 0.303i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.429 - 0.744i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.50T + 31T^{2} \) |
| 37 | \( 1 + 9.38iT - 37T^{2} \) |
| 41 | \( 1 + (2.06 - 3.57i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (8.76 + 5.06i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-9.10 + 5.25i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.31 - 2.49i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.12 + 5.41i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.27 - 3.94i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.48 + 2.59i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.58 - 11.4i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.7 - 6.18i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.98 - 5.16i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 + (7.98 + 13.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.4 - 8.35i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.20487091291818503500621720721, −9.161768380935054558949024994594, −8.611801802029341730650112083474, −7.15635071065666071244093288832, −6.42795368131141651960476886138, −5.61029500777082370715814069938, −4.85038267012175280933808789637, −3.71992912438363749034013485980, −2.29245236740990240508700181069, −0.859770131226060018981051926659,
1.72557663941257551390393902325, 2.86573963849753881525397971391, 4.01054691090270047640775674489, 4.95214094320165282630573216886, 5.93126927734920980111807196598, 6.70859048245634069294390694342, 8.103617189756088208019243286792, 8.634977154604726313123055956014, 9.383246153031325231870925402640, 10.30212692704647002337228564268