| L(s) = 1 | + (0.910 − 1.27i)2-s + (−0.235 + 0.971i)3-s + (−0.151 − 0.438i)4-s + (0.0858 − 1.80i)5-s + (1.02 + 1.18i)6-s + (−0.649 − 2.56i)7-s + (2.31 + 0.679i)8-s + (−0.888 − 0.458i)9-s + (−2.22 − 1.75i)10-s + (−3.08 − 4.33i)11-s + (0.461 − 0.0440i)12-s + (−0.0994 + 0.691i)13-s + (−3.87 − 1.50i)14-s + (1.73 + 0.508i)15-s + (3.70 − 2.91i)16-s + (1.31 − 0.253i)17-s + ⋯ |
| L(s) = 1 | + (0.643 − 0.904i)2-s + (−0.136 + 0.561i)3-s + (−0.0758 − 0.219i)4-s + (0.0383 − 0.805i)5-s + (0.419 + 0.484i)6-s + (−0.245 − 0.969i)7-s + (0.817 + 0.240i)8-s + (−0.296 − 0.152i)9-s + (−0.703 − 0.553i)10-s + (−0.930 − 1.30i)11-s + (0.133 − 0.0127i)12-s + (−0.0275 + 0.191i)13-s + (−1.03 − 0.402i)14-s + (0.446 + 0.131i)15-s + (0.926 − 0.728i)16-s + (0.319 − 0.0615i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26147 - 1.40106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26147 - 1.40106i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.235 - 0.971i)T \) |
| 7 | \( 1 + (0.649 + 2.56i)T \) |
| 23 | \( 1 + (-1.25 + 4.62i)T \) |
| good | 2 | \( 1 + (-0.910 + 1.27i)T + (-0.654 - 1.89i)T^{2} \) |
| 5 | \( 1 + (-0.0858 + 1.80i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (3.08 + 4.33i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (0.0994 - 0.691i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.31 + 0.253i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (-6.91 - 1.33i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (-0.729 - 0.841i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (5.80 - 5.53i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (2.54 + 1.31i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (5.87 + 3.77i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (6.31 - 1.85i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-3.25 - 5.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.87 - 2.75i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (-6.70 - 5.27i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (-2.04 - 8.41i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (7.62 + 0.728i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (1.77 + 3.89i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.53 - 7.33i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (-12.7 + 5.08i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (-4.03 + 2.59i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (0.477 + 0.455i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (-7.17 - 4.61i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.64425792592636074781073655633, −10.37431816353188060169654083079, −9.094579569055002076255829955048, −8.152367911759651515653460686885, −7.13313446160093870848016376695, −5.50175246676108982153232967071, −4.87678758962080066830140448591, −3.72105210986933681273496710007, −2.99629289372660802146171795056, −1.02997531899605607837870923026,
2.02271644525977287331848228002, 3.30807374417006936532730274960, 5.12642604181716864637006416987, 5.50194808403647425693816345197, 6.71185130249747590037829674719, 7.26281514308957080202702934873, 8.036009161713091852624586680530, 9.546827298461558180094482002499, 10.26078241897223096776974311259, 11.38065491328687348686190048684
