L(s) = 1 | + (0.707 − 1.22i)2-s + (−1.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s + (1.93 + 1.11i)5-s + 2.44i·6-s + (4.24 + 5.56i)7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (2.73 − 1.58i)10-s + (5.42 + 9.40i)11-s + (2.99 + 1.73i)12-s − 0.772i·13-s + (9.81 − 1.26i)14-s − 3.87·15-s + (−2.00 + 3.46i)16-s + (16.7 − 9.68i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + 0.408i·6-s + (0.606 + 0.795i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.273 − 0.158i)10-s + (0.493 + 0.854i)11-s + (0.249 + 0.144i)12-s − 0.0593i·13-s + (0.701 − 0.0902i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (0.986 − 0.569i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00120i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.00120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.76957 + 0.00106236i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.76957 + 0.00106236i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 5 | \( 1 + (-1.93 - 1.11i)T \) |
| 7 | \( 1 + (-4.24 - 5.56i)T \) |
good | 11 | \( 1 + (-5.42 - 9.40i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 0.772iT - 169T^{2} \) |
| 17 | \( 1 + (-16.7 + 9.68i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-22.5 - 13.0i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-6.84 + 11.8i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 6.99T + 841T^{2} \) |
| 31 | \( 1 + (-22.7 + 13.1i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (32.3 - 55.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 5.54iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (19.5 + 11.3i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (37.2 + 64.5i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-96.6 + 55.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (46.9 + 27.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-22.1 - 38.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 31.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (92.6 - 53.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (14.8 - 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 15.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (31.8 + 18.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 134. iT - 9.40e3T^{2} \) |
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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
−11.84594762059159049571916895603, −11.54940686394528545639110170556, −10.07070529735011587306228716315, −9.681914781975932205455140365994, −8.286392493409147733149985807021, −6.80010788613765180166400658380, −5.53743677976992720501840635130, −4.79104181325984302140693570932, −3.19363976635606995269604861580, −1.58976726596958184663327864551,
1.15902809124987229929026598409, 3.52910450614419869575977570498, 4.95293590409190754437598790517, 5.82798294801867947246358509485, 6.98613285381387366560009887803, 7.87740357138089137737258910959, 9.016516862853258934331787772552, 10.27998405640106200238493723375, 11.34144745778724143323375563773, 12.16121538786538754856410874346