| L(s) = 1 | + (−0.629 − 1.67i)2-s + (−1.66 + 0.491i)3-s + (−0.908 + 0.793i)4-s + (1.40 + 1.93i)5-s + (1.86 + 2.47i)6-s + (2.34 − 1.54i)7-s + (−1.25 − 0.673i)8-s + (2.51 − 1.63i)9-s + (2.36 − 3.57i)10-s + (−0.286 + 0.300i)11-s + (1.11 − 1.76i)12-s + (3.89 − 3.72i)13-s + (−4.06 − 2.95i)14-s + (−3.28 − 2.52i)15-s + (−0.665 + 4.91i)16-s + (−0.620 − 1.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.444 − 1.18i)2-s + (−0.958 + 0.283i)3-s + (−0.454 + 0.396i)4-s + (0.629 + 0.865i)5-s + (0.763 + 1.01i)6-s + (0.884 − 0.583i)7-s + (−0.442 − 0.238i)8-s + (0.838 − 0.544i)9-s + (0.746 − 1.13i)10-s + (−0.0864 + 0.0904i)11-s + (0.322 − 0.509i)12-s + (1.07 − 1.03i)13-s + (−1.08 − 0.788i)14-s + (−0.849 − 0.651i)15-s + (−0.166 + 1.22i)16-s + (−0.150 − 0.463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00350 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00350 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.621892 - 0.624074i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.621892 - 0.624074i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.66 - 0.491i)T \) |
| 71 | \( 1 + (1.38 - 8.31i)T \) |
| good | 2 | \( 1 + (0.629 + 1.67i)T + (-1.50 + 1.31i)T^{2} \) |
| 5 | \( 1 + (-1.40 - 1.93i)T + (-1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-2.34 + 1.54i)T + (2.75 - 6.43i)T^{2} \) |
| 11 | \( 1 + (0.286 - 0.300i)T + (-0.493 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-3.89 + 3.72i)T + (0.583 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.620 + 1.90i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.77 + 0.160i)T + (18.6 + 3.39i)T^{2} \) |
| 23 | \( 1 + (-1.96 + 2.45i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (-8.36 + 3.57i)T + (20.0 - 20.9i)T^{2} \) |
| 31 | \( 1 + (4.74 - 0.642i)T + (29.8 - 8.24i)T^{2} \) |
| 37 | \( 1 + (-5.40 - 6.78i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + (6.21 - 2.99i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-4.85 - 1.33i)T + (36.9 + 22.0i)T^{2} \) |
| 47 | \( 1 + (-4.69 + 2.80i)T + (22.2 - 41.3i)T^{2} \) |
| 53 | \( 1 + (1.78 + 1.55i)T + (7.11 + 52.5i)T^{2} \) |
| 59 | \( 1 + (12.5 - 2.27i)T + (55.2 - 20.7i)T^{2} \) |
| 61 | \( 1 + (-4.07 - 2.69i)T + (23.9 + 56.0i)T^{2} \) |
| 67 | \( 1 + (4.12 + 4.71i)T + (-8.99 + 66.3i)T^{2} \) |
| 73 | \( 1 + (12.0 - 4.52i)T + (54.9 - 48.0i)T^{2} \) |
| 79 | \( 1 + (6.35 - 11.8i)T + (-43.5 - 65.9i)T^{2} \) |
| 83 | \( 1 + (-1.25 - 6.93i)T + (-77.7 + 29.1i)T^{2} \) |
| 89 | \( 1 + (0.779 - 0.892i)T + (-11.9 - 88.1i)T^{2} \) |
| 97 | \( 1 + (7.80 - 16.1i)T + (-60.4 - 75.8i)T^{2} \) |
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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
−11.67287276506943860338840058327, −10.86265352162533218382868535965, −10.55210204815642916570599368324, −9.757314185612774157802057755559, −8.354562179976078618766877440121, −6.80358484321281960046384362261, −5.88557090343917610027803766273, −4.39453249035729870743711449908, −2.87384702350920106137497038473, −1.15656557334084833334663955672,
1.63920733076964317956235701396, 4.65718458276695601297164713656, 5.63361619495108375308292480801, 6.29656669218935058852646654664, 7.46545040375982238092549634752, 8.630053597553090333805640957296, 9.143730840312169587512666024027, 10.74482826022056696842267413584, 11.65079349214811795369800269766, 12.53390788587966205148355441713
