| L(s) = 1 | + (−0.325 + 1.00i)2-s + (−1.99 − 4.79i)3-s + (5.57 + 4.05i)4-s + (11.6 − 3.79i)5-s + (5.44 − 0.433i)6-s + (13.1 − 18.1i)7-s + (−12.6 + 9.20i)8-s + (−19.0 + 19.1i)9-s + 12.9i·10-s + (12.4 + 34.2i)11-s + (8.33 − 34.8i)12-s + (−75.4 − 24.5i)13-s + (13.8 + 19.0i)14-s + (−41.5 − 48.5i)15-s + (11.9 + 36.7i)16-s + (−10.2 − 31.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.114 + 0.353i)2-s + (−0.383 − 0.923i)3-s + (0.697 + 0.506i)4-s + (1.04 − 0.339i)5-s + (0.370 − 0.0294i)6-s + (0.710 − 0.978i)7-s + (−0.560 + 0.406i)8-s + (−0.705 + 0.708i)9-s + 0.408i·10-s + (0.341 + 0.939i)11-s + (0.200 − 0.838i)12-s + (−1.60 − 0.522i)13-s + (0.264 + 0.363i)14-s + (−0.714 − 0.835i)15-s + (0.186 + 0.574i)16-s + (−0.146 − 0.452i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.33052 - 0.148351i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33052 - 0.148351i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.99 + 4.79i)T \) |
| 11 | \( 1 + (-12.4 - 34.2i)T \) |
| good | 2 | \( 1 + (0.325 - 1.00i)T + (-6.47 - 4.70i)T^{2} \) |
| 5 | \( 1 + (-11.6 + 3.79i)T + (101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (-13.1 + 18.1i)T + (-105. - 326. i)T^{2} \) |
| 13 | \( 1 + (75.4 + 24.5i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (10.2 + 31.6i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-21.2 - 29.2i)T + (-2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 - 134. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (87.6 + 63.7i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-9.67 + 29.7i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (6.44 + 4.68i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-6.01 + 4.37i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 74.3iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (301. + 414. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-327. - 106. i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-344. + 474. i)T + (-6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (243. - 79.2i)T + (1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 - 664.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (3.67 - 1.19i)T + (2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-118. + 163. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-172. - 56.1i)T + (3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-235. - 725. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 311. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-231. + 711. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−16.76173848830080563194025517302, −14.89841122601236924322276814358, −13.69056144541552981505052669824, −12.52594380654916071362409918560, −11.45575212221694846679837280682, −9.840014330274510963645079216776, −7.77522719875785629844970477743, −7.01579765798349791687471958227, −5.33108502732454187633535799635, −1.96132791098703815346136095074,
2.44988082736689128110538046338, 5.25036182598543519404626730634, 6.37937157842539733862808660911, 9.026174819054473546515645799980, 10.06808735074495929948828603153, 11.15755381082551890764982589623, 12.12264296451578332290305372456, 14.38452429175596064322920683074, 14.90672508276371195430477908381, 16.29167977955102997607734139667
