L(s) = 1 | − 4.48·2-s + (−2.57 − 4.51i)3-s + 12.1·4-s + 12.2i·5-s + (11.5 + 20.2i)6-s + 14.5i·7-s − 18.6·8-s + (−13.7 + 23.2i)9-s − 55.0i·10-s + (27.6 + 23.8i)11-s + (−31.3 − 54.8i)12-s − 14.5i·13-s − 65.2i·14-s + (55.3 − 31.5i)15-s − 13.5·16-s − 92.5·17-s + ⋯ |
L(s) = 1 | − 1.58·2-s + (−0.495 − 0.868i)3-s + 1.51·4-s + 1.09i·5-s + (0.786 + 1.37i)6-s + 0.784i·7-s − 0.823·8-s + (−0.508 + 0.861i)9-s − 1.74i·10-s + (0.756 + 0.653i)11-s + (−0.753 − 1.31i)12-s − 0.310i·13-s − 1.24i·14-s + (0.952 − 0.543i)15-s − 0.211·16-s − 1.31·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.308526 + 0.253965i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.308526 + 0.253965i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.57 + 4.51i)T \) |
| 11 | \( 1 + (-27.6 - 23.8i)T \) |
good | 2 | \( 1 + 4.48T + 8T^{2} \) |
| 5 | \( 1 - 12.2iT - 125T^{2} \) |
| 7 | \( 1 - 14.5iT - 343T^{2} \) |
| 13 | \( 1 + 14.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 92.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 162. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 23.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 52.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 82.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 276.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 175.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 353. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 387. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 79.9iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 291. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 147. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 201.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 424. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 461. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 122. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 151.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 764. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 218.T + 9.12e5T^{2} \) |
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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
−17.05346347691746239267423018262, −15.63805655997744073784273275991, −14.26929265180111061970405660315, −12.37467382753750993751024310634, −11.24818902548211555943800457635, −10.19257545094308088785843043310, −8.645392196584240234297141862038, −7.31888452655886280069096564305, −6.28504213408638895402795814910, −2.04747580549261023686850057389,
0.63875107001131564116439778891, 4.55066301893890813421406617855, 6.78024524585597075763815532555, 8.752596983911148175662511435827, 9.275850971674146396115462944911, 10.73119967856503054379055621483, 11.57997276072087298477069379164, 13.51124677549888252956281579545, 15.51574928751758935061387709003, 16.43581046438449545187848607129