L(s) = 1 | + (1.76 − 1.01i)5-s − 3.69·7-s + 0.605i·11-s + (2.65 + 1.53i)13-s + (2.40 − 1.39i)17-s + (−0.780 + 4.28i)19-s + (2.23 + 1.29i)23-s + (−0.422 + 0.732i)25-s + (−1.70 + 2.95i)29-s − 4.42i·31-s + (−6.52 + 3.76i)35-s + 3.34i·37-s + (−1.50 − 2.60i)41-s + (−0.562 − 0.974i)43-s + (3.05 + 1.76i)47-s + ⋯ |
L(s) = 1 | + (0.789 − 0.455i)5-s − 1.39·7-s + 0.182i·11-s + (0.735 + 0.424i)13-s + (0.584 − 0.337i)17-s + (−0.179 + 0.983i)19-s + (0.466 + 0.269i)23-s + (−0.0845 + 0.146i)25-s + (−0.316 + 0.548i)29-s − 0.795i·31-s + (−1.10 + 0.637i)35-s + 0.550i·37-s + (−0.234 − 0.406i)41-s + (−0.0858 − 0.148i)43-s + (0.445 + 0.257i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.682579821\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.682579821\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (0.780 - 4.28i)T \) |
good | 5 | \( 1 + (-1.76 + 1.01i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 - 0.605iT - 11T^{2} \) |
| 13 | \( 1 + (-2.65 - 1.53i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.40 + 1.39i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.23 - 1.29i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.70 - 2.95i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.42iT - 31T^{2} \) |
| 37 | \( 1 - 3.34iT - 37T^{2} \) |
| 41 | \( 1 + (1.50 + 2.60i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.562 + 0.974i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.05 - 1.76i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.984 - 1.70i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.48 - 6.03i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.98 + 8.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.324 + 0.187i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.03 - 13.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.09 - 14.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.44 + 4.30i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.17iT - 83T^{2} \) |
| 89 | \( 1 + (3.91 - 6.77i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.57 + 4.95i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.066250291678635537539005081782, −8.283974934229984026236605986230, −7.25760285011299612609401912550, −6.53146621238333519183845534186, −5.83677603639828803637233737364, −5.24753577961798060804930233519, −3.97227528182729958349507093762, −3.32468002270257937807817397103, −2.17922020019422260274479372234, −1.06086500345307531070965325055,
0.62925482168475008156541288018, 2.14046628142084896313844709714, 3.08523542311546266705720314030, 3.67167984238686750140635340428, 4.95613076854116599692556365850, 5.92670551727660438632168683180, 6.35737808116190535311594354642, 7.00756905867464729143801115649, 8.008962813125077333981526595972, 8.935382722696306519137354199030