| L(s) = 1 | + (−0.0404 − 0.124i)2-s + (−2.42 − 1.76i)3-s + (6.45 − 4.69i)4-s + (2.06 − 6.36i)5-s + (−0.121 + 0.373i)6-s + (11.6 − 8.43i)7-s + (−1.69 − 1.22i)8-s + (2.78 + 8.55i)9-s − 0.875·10-s + (−28.3 + 22.9i)11-s − 23.9·12-s + (10.8 + 33.2i)13-s + (−1.51 − 1.10i)14-s + (−16.2 + 11.7i)15-s + (19.6 − 60.4i)16-s + (−21.6 + 66.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0142 − 0.0439i)2-s + (−0.467 − 0.339i)3-s + (0.807 − 0.586i)4-s + (0.184 − 0.569i)5-s + (−0.00825 + 0.0254i)6-s + (0.626 − 0.455i)7-s + (−0.0747 − 0.0543i)8-s + (0.103 + 0.317i)9-s − 0.0276·10-s + (−0.777 + 0.628i)11-s − 0.576·12-s + (0.230 + 0.710i)13-s + (−0.0289 − 0.0210i)14-s + (−0.279 + 0.203i)15-s + (0.307 − 0.944i)16-s + (−0.308 + 0.949i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.14659 - 0.580377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14659 - 0.580377i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.42 + 1.76i)T \) |
| 11 | \( 1 + (28.3 - 22.9i)T \) |
| good | 2 | \( 1 + (0.0404 + 0.124i)T + (-6.47 + 4.70i)T^{2} \) |
| 5 | \( 1 + (-2.06 + 6.36i)T + (-101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (-11.6 + 8.43i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (-10.8 - 33.2i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (21.6 - 66.5i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-45.0 - 32.7i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + 43.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-168. + 122. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (38.3 + 117. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (316. - 230. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-340. - 247. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 410.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (177. + 128. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (109. + 336. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (2.98 - 2.16i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-37.7 + 116. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + 219.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-332. + 1.02e3i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-592. + 430. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (85.3 + 262. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (117. - 361. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-242. - 746. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−16.14825450962444582730532441880, −14.95041464787631140021333327428, −13.58670305180816450105208925020, −12.22433293518315047771638756707, −11.09549057166904987688725473134, −9.970857651058490497401926322868, −7.955955324313110903886441370561, −6.48380120995525336123793502370, −4.95055038512853184718436171382, −1.63861996280661918398598882024,
2.91338334474154452240175044150, 5.40469474319773766701555659949, 6.99300540714576538332452770204, 8.478831329337010334836303467898, 10.50411846952161959459010329825, 11.31094759949897228353438705863, 12.50026046935347462117926622132, 14.13435310077963512578529977679, 15.58979036380684981491719510795, 16.14428258359993272818418497355
