L(s) = 1 | + (1.50 − 1.09i)2-s + (0.309 + 0.951i)3-s + (0.453 − 1.39i)4-s + (1.50 + 1.09i)6-s + (−0.812 + 2.50i)7-s + (0.306 + 0.943i)8-s + (−0.809 + 0.587i)9-s + (1.77 + 2.80i)11-s + 1.46·12-s + (−5.24 + 3.80i)13-s + (1.51 + 4.65i)14-s + (3.86 + 2.81i)16-s + (−3.82 − 2.77i)17-s + (−0.575 + 1.77i)18-s + (0.164 + 0.506i)19-s + ⋯ |
L(s) = 1 | + (1.06 − 0.773i)2-s + (0.178 + 0.549i)3-s + (0.226 − 0.697i)4-s + (0.614 + 0.446i)6-s + (−0.307 + 0.945i)7-s + (0.108 + 0.333i)8-s + (−0.269 + 0.195i)9-s + (0.534 + 0.845i)11-s + 0.423·12-s + (−1.45 + 1.05i)13-s + (0.404 + 1.24i)14-s + (0.967 + 0.702i)16-s + (−0.926 − 0.673i)17-s + (−0.135 + 0.417i)18-s + (0.0377 + 0.116i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.28960 + 1.04221i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.28960 + 1.04221i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-1.77 - 2.80i)T \) |
good | 2 | \( 1 + (-1.50 + 1.09i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (0.812 - 2.50i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (5.24 - 3.80i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.82 + 2.77i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.164 - 0.506i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 + (-3.28 + 10.1i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.154 - 0.112i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.30 + 4.02i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.187 + 0.575i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 + (-2.72 - 8.39i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.41 + 3.93i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.83 - 5.63i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.60 - 1.89i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 5.22T + 67T^{2} \) |
| 71 | \( 1 + (-2.21 - 1.61i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.84 + 14.9i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.979 - 0.711i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.77 - 2.74i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 6.40T + 89T^{2} \) |
| 97 | \( 1 + (5.92 - 4.30i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.48764833367691103591562201307, −9.325838054361912316859699262060, −9.217710782831004322792720249858, −7.75056945066388958945685339483, −6.68918908679024194416212936770, −5.56847871154129978267618762525, −4.58919231493932081994718865387, −4.19806566875735060191336296178, −2.66898585899728789429365750946, −2.25612105876794567275857140970,
0.872025677249746738072860978454, 2.87233964116577775348455802586, 3.80579340599987030129738980257, 4.85912170601302448907886571082, 5.71151921405921593743545508460, 6.81618486687405158815337366366, 7.07033947129149766441176392853, 8.098398331850279024308307632585, 9.114871554193384225253724061398, 10.24006510769264008848914452754