L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − i·5-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (−0.5 − 0.866i)10-s + (−2.36 + 1.36i)11-s + (−2.59 − 2.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (1.13 − 1.96i)17-s + (−4.73 − 2.73i)19-s + (−0.866 − 0.499i)20-s + (−1.36 + 2.36i)22-s + (−2.36 − 4.09i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (−0.158 − 0.273i)10-s + (−0.713 + 0.411i)11-s + (−0.720 − 0.693i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (0.275 − 0.476i)17-s + (−1.08 − 0.626i)19-s + (−0.193 − 0.111i)20-s + (−0.291 + 0.504i)22-s + (−0.493 − 0.854i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.758890874\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.758890874\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (2.59 + 2.5i)T \) |
good | 5 | \( 1 + iT - 5T^{2} \) |
| 11 | \( 1 + (2.36 - 1.36i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.13 + 1.96i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.73 + 2.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.36 + 4.09i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.73iT - 31T^{2} \) |
| 37 | \( 1 + (-4.5 + 2.59i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.86 + 2.23i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.901 - 1.56i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 10.9iT - 47T^{2} \) |
| 53 | \( 1 + 8.46T + 53T^{2} \) |
| 59 | \( 1 + (6.29 + 3.63i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.13 + 7.16i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.90 + 2.83i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-12 - 6.92i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.39iT - 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 14.1iT - 83T^{2} \) |
| 89 | \( 1 + (-8.19 + 4.73i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.26 + 0.732i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.269756086298253180629597852018, −8.145675314564918675193541421485, −7.61427900073444027936064825597, −6.49540048774262671200152617806, −5.62860401504522824565482421575, −4.79323834364078220596117714792, −4.28064700303054041997013428516, −2.81382713616734582485322327063, −2.17918881391677020688585748484, −0.51320553143120878207139876125,
1.76742628410608999798394077113, 2.89795987227696750433849554760, 3.83790990263052603778811023615, 4.78018447693961449369066234724, 5.56238110607728963532683901455, 6.48702976482005640894131298981, 7.16232769173850827032317162233, 8.001374690956146307488222139034, 8.640969362158583743327344557666, 9.755412547060618272463833644221