| L(s) = 1 | + (2.59 + 15.7i)2-s + (−23.9 − 7.77i)3-s + (−242. + 81.8i)4-s + (443. − 440. i)5-s + (60.7 − 397. i)6-s + 2.97e3i·7-s + (−1.92e3 − 3.61e3i)8-s + (−4.79e3 − 3.48e3i)9-s + (8.10e3 + 5.85e3i)10-s + (1.57e3 + 2.16e3i)11-s + (6.43e3 − 71.7i)12-s + (−7.22e3 − 5.25e3i)13-s + (−4.70e4 + 7.71e3i)14-s + (−1.40e4 + 7.09e3i)15-s + (5.21e4 − 3.96e4i)16-s + (−5.40e3 − 1.66e4i)17-s + ⋯ |
| L(s) = 1 | + (0.161 + 0.986i)2-s + (−0.295 − 0.0959i)3-s + (−0.947 + 0.319i)4-s + (0.708 − 0.705i)5-s + (0.0468 − 0.306i)6-s + 1.24i·7-s + (−0.468 − 0.883i)8-s + (−0.731 − 0.531i)9-s + (0.810 + 0.585i)10-s + (0.107 + 0.147i)11-s + (0.310 − 0.00346i)12-s + (−0.253 − 0.183i)13-s + (−1.22 + 0.200i)14-s + (−0.277 + 0.140i)15-s + (0.795 − 0.605i)16-s + (−0.0647 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.688i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.724 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.63089 + 0.651243i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63089 + 0.651243i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.59 - 15.7i)T \) |
| 5 | \( 1 + (-443. + 440. i)T \) |
| good | 3 | \( 1 + (23.9 + 7.77i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 - 2.97e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-1.57e3 - 2.16e3i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (7.22e3 + 5.25e3i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (5.40e3 + 1.66e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-7.23e4 + 2.35e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (5.55e4 + 7.65e4i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (-1.05e5 + 3.25e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-1.08e6 + 3.52e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-1.80e6 - 1.31e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-2.33e6 - 1.69e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 2.10e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-5.34e5 - 1.73e5i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (2.11e6 - 6.51e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-5.55e6 + 7.64e6i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (1.62e6 - 1.18e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (1.60e7 - 5.22e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-3.07e7 - 9.99e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (2.43e7 - 1.76e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-5.19e7 - 1.68e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-8.13e7 + 2.64e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-8.23e7 + 5.98e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (-2.82e7 + 8.69e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−12.47520991109516289136746735038, −11.76791850611254139261957122695, −9.739772953061163591978392247478, −9.016455812779552849052047124421, −8.070728074857096490782705459312, −6.35900934310171788741576228169, −5.70839350397331864294922519074, −4.69650702198571462911957642513, −2.71893856804809535269360643297, −0.70236349524337830013115182617,
0.857777213823157764582255979983, 2.32863719534762259517669655184, 3.59504728964906818091242155459, 4.98643823167678601019110928212, 6.23209453425074655861756943035, 7.75189469038348489148944908126, 9.302788485474075671297306436197, 10.38209090355414303197677438301, 10.87473115166332475404461338850, 11.92052005834043381834977045142
