L(s) = 1 | + (−2.69 − 0.856i)2-s + (6.53 + 4.61i)4-s − 10.3i·5-s + (−16.6 − 8.15i)7-s + (−13.6 − 18.0i)8-s + (−8.83 + 27.7i)10-s − 18.0i·11-s + 49.0i·13-s + (37.8 + 36.2i)14-s + (21.3 + 60.3i)16-s − 46.6i·17-s − 48.8·19-s + (47.6 − 67.3i)20-s + (−15.4 + 48.6i)22-s + 33.7i·23-s + ⋯ |
L(s) = 1 | + (−0.952 − 0.302i)2-s + (0.816 + 0.577i)4-s − 0.921i·5-s + (−0.897 − 0.440i)7-s + (−0.603 − 0.797i)8-s + (−0.279 + 0.878i)10-s − 0.494i·11-s + 1.04i·13-s + (0.722 + 0.691i)14-s + (0.332 + 0.942i)16-s − 0.666i·17-s − 0.589·19-s + (0.532 − 0.752i)20-s + (−0.149 + 0.471i)22-s + 0.306i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.07441720474\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07441720474\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.69 + 0.856i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (16.6 + 8.15i)T \) |
good | 5 | \( 1 + 10.3iT - 125T^{2} \) |
| 11 | \( 1 + 18.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 49.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 46.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 33.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 48.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 409. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 470. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 548.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 203.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 717.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 493. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 240. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 995. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 790. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 214. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 885.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 67.3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 934. iT - 9.12e5T^{2} \) |
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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
−11.73461479127970249755419389423, −10.94285894160112224621503840811, −9.697072091635274813004313537266, −9.211927561015032994150486917004, −8.238658348507665207227062797318, −7.08953080080408845597821909307, −6.13736468900251889080548316007, −4.43243955309962168699147288131, −3.07660890104141205424189159358, −1.37006523864020470581703312032,
0.04049908989939986391806504233, 2.18386791455983477343098432285, 3.37844139646684023526086716010, 5.52018501100598008295008638276, 6.48182985861608770478091579220, 7.26414325575004781438162617878, 8.389464399710484383746599584413, 9.387300475687373133524877863453, 10.37791716894575002590748134297, 10.80501527797336398510055813586