L(s) = 1 | + (0.841 + 0.540i)2-s + (−0.473 − 3.29i)3-s + (0.415 + 0.909i)4-s + (−2.68 − 0.789i)5-s + (1.38 − 3.02i)6-s + (0.654 − 0.755i)7-s + (−0.142 + 0.989i)8-s + (−7.74 + 2.27i)9-s + (−1.83 − 2.11i)10-s + (2.03 − 1.31i)11-s + (2.79 − 1.79i)12-s + (−1.56 − 1.80i)13-s + (0.959 − 0.281i)14-s + (−1.32 + 9.22i)15-s + (−0.654 + 0.755i)16-s + (−1.04 + 2.28i)17-s + ⋯ |
L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.273 − 1.90i)3-s + (0.207 + 0.454i)4-s + (−1.20 − 0.352i)5-s + (0.564 − 1.23i)6-s + (0.247 − 0.285i)7-s + (−0.0503 + 0.349i)8-s + (−2.58 + 0.758i)9-s + (−0.579 − 0.669i)10-s + (0.614 − 0.395i)11-s + (0.808 − 0.519i)12-s + (−0.434 − 0.500i)13-s + (0.256 − 0.0752i)14-s + (−0.342 + 2.38i)15-s + (−0.163 + 0.188i)16-s + (−0.252 + 0.552i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.415096 - 1.03375i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.415096 - 1.03375i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 7 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (-4.38 - 1.95i)T \) |
good | 3 | \( 1 + (0.473 + 3.29i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (2.68 + 0.789i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-2.03 + 1.31i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (1.56 + 1.80i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.04 - 2.28i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (2.73 + 5.98i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.32 + 5.09i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.688 + 4.79i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (2.54 - 0.746i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-10.2 - 3.01i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.540 + 3.75i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 4.68T + 47T^{2} \) |
| 53 | \( 1 + (-9.27 + 10.7i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-7.89 - 9.11i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.362 + 2.51i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (3.50 + 2.25i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (6.22 + 4.00i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.08 - 2.36i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-5.38 - 6.21i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (16.2 - 4.76i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.504 - 3.50i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (8.20 + 2.41i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.58954154621690801659285547080, −11.04080617241080366229045647022, −8.769187503459515583353734418121, −8.070637490696218834711385758834, −7.30776339288172900283143035089, −6.61818402990498079620838544448, −5.49978095615432035725834778708, −4.16106046986932816730758674966, −2.57807607887822432872376093281, −0.68721493067723082604168090615,
2.95280061584083501559641372538, 4.03922329530302761670472520252, 4.52932337475570086839636153990, 5.64226905880142801135159909931, 7.01027205444679334981430979378, 8.555435899319710970495061618176, 9.365513252342313715266373725252, 10.43385206199337235110618848481, 11.00156481083629589128561242569, 11.83784701922926436672492728093