| 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
−16.14428258359993272818418497355, −15.58979036380684981491719510795, −14.13435310077963512578529977679, −12.50026046935347462117926622132, −11.31094759949897228353438705863, −10.50411846952161959459010329825, −8.478831329337010334836303467898, −6.99300540714576538332452770204, −5.40469474319773766701555659949, −2.91338334474154452240175044150,
1.63861996280661918398598882024, 4.95055038512853184718436171382, 6.48380120995525336123793502370, 7.955955324313110903886441370561, 9.970857651058490497401926322868, 11.09549057166904987688725473134, 12.22433293518315047771638756707, 13.58670305180816450105208925020, 14.95041464787631140021333327428, 16.14825450962444582730532441880
