| L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.741 − 1.28i)5-s + (−1.48 + 2.19i)7-s + 0.999i·8-s + (1.28 − 0.741i)10-s + (0.866 − 0.5i)11-s − 2.44i·13-s + (−2.38 + 1.15i)14-s + (−0.5 + 0.866i)16-s + (−0.182 − 0.315i)17-s + (7.06 + 4.07i)19-s + 1.48·20-s + 0.999·22-s + (5.18 + 2.99i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.331 − 0.574i)5-s + (−0.560 + 0.827i)7-s + 0.353i·8-s + (0.405 − 0.234i)10-s + (0.261 − 0.150i)11-s − 0.679i·13-s + (−0.636 + 0.308i)14-s + (−0.125 + 0.216i)16-s + (−0.0441 − 0.0765i)17-s + (1.62 + 0.935i)19-s + 0.331·20-s + 0.213·22-s + (1.08 + 0.623i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.502934616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.502934616\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.48 - 2.19i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| good | 5 | \( 1 + (-0.741 + 1.28i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (0.182 + 0.315i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-7.06 - 4.07i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.18 - 2.99i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.332iT - 29T^{2} \) |
| 31 | \( 1 + (-0.752 + 0.434i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.64 - 8.04i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9.38T + 41T^{2} \) |
| 43 | \( 1 + 2.94T + 43T^{2} \) |
| 47 | \( 1 + (0.637 - 1.10i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.65 + 2.68i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.43 - 7.68i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.28 + 4.20i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.08 + 8.79i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.07iT - 71T^{2} \) |
| 73 | \( 1 + (7.86 - 4.54i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.00 + 5.21i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + (-0.459 + 0.796i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 1.79iT - 97T^{2} \) |
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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
−9.487119762613400658446757855286, −8.967622198923547630215429433378, −8.011453985439188058499446645674, −7.20019546747157297170420270925, −6.17657850083899196323243757083, −5.48494344855047988125820451381, −4.95339695684235041875694511420, −3.53231458111168320101523058175, −2.87661057309070680041859703242, −1.33662354323557497185627644563,
0.966540208665610387484184556035, 2.46362859258588318191638001736, 3.30645694263501414407191431435, 4.25408156918623769616882735816, 5.14346264167712771208547792527, 6.23732152754843106814116191147, 6.94581610870495287229324351801, 7.43494636177320385006816349995, 8.969722394836420029575828386081, 9.536616335669846694486338120902
