| L(s) = 1 | + (−0.587 + 0.809i)2-s + (−2.74 + 1.21i)3-s + (0.927 + 2.85i)4-s + (3.88 + 5.35i)5-s + (0.633 − 2.93i)6-s + (−2.73 − 8.42i)7-s + (−6.65 − 2.16i)8-s + (6.06 − 6.64i)9-s − 6.61·10-s + (10.8 + 1.76i)11-s + (−6 − 6.70i)12-s + (10.7 + 7.77i)13-s + (8.42 + 2.73i)14-s + (−17.1 − 9.98i)15-s + (−4.04 + 2.93i)16-s + (−3.52 − 4.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.404i)2-s + (−0.914 + 0.403i)3-s + (0.231 + 0.713i)4-s + (0.777 + 1.07i)5-s + (0.105 − 0.488i)6-s + (−0.390 − 1.20i)7-s + (−0.832 − 0.270i)8-s + (0.674 − 0.738i)9-s − 0.661·10-s + (0.987 + 0.160i)11-s + (−0.5 − 0.559i)12-s + (0.823 + 0.598i)13-s + (0.601 + 0.195i)14-s + (−1.14 − 0.665i)15-s + (−0.252 + 0.183i)16-s + (−0.207 − 0.285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0358 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0358 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.565847 + 0.545906i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.565847 + 0.545906i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.74 - 1.21i)T \) |
| 11 | \( 1 + (-10.8 - 1.76i)T \) |
| good | 2 | \( 1 + (0.587 - 0.809i)T + (-1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (-3.88 - 5.35i)T + (-7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (2.73 + 8.42i)T + (-39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (-10.7 - 7.77i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (3.52 + 4.85i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (-2.81 + 8.67i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + 17.5iT - 529T^{2} \) |
| 29 | \( 1 + (25.1 - 8.15i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-5.01 - 3.64i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (6.05 + 18.6i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-5.32 - 1.72i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + 26.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (16.8 + 5.47i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-13.0 + 18.0i)T + (-868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (20.8 - 6.77i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-75.5 + 54.8i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + 76.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-38.6 - 53.2i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-4.64 - 14.2i)T + (-4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (95.5 + 69.3i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-65.3 - 89.9i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 + 97.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-98.0 - 71.2i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−16.87208356996980861618585889551, −16.03505293797734356073107180560, −14.53450646991226959471450683725, −13.21403714780659579519572453169, −11.60349671359323166494485380744, −10.58345353570825658069440377990, −9.316486976126677596717067577431, −6.96920471485144085942736165911, −6.45483134936640174729410204939, −3.80793319045269849458489798362,
1.51417195775183990309847074135, 5.49599414995704841251442265496, 6.15902132055586612423830608906, 8.812610613004002191852516583785, 9.855614359660789335607361500839, 11.37174336602229951466797209561, 12.34100384015094031673448558316, 13.45947084875904749960806627865, 15.25422009903055760342204296500, 16.39769041180243344186230044857
