| L(s) = 1 | + (1.38 − 4.25i)2-s + (−2.42 + 1.76i)3-s + (−9.70 − 7.05i)4-s + (−5.80 − 17.8i)5-s + (4.14 + 12.7i)6-s + (23.2 + 16.8i)7-s + (−14.4 + 10.5i)8-s + (2.78 − 8.55i)9-s − 84.0·10-s + (27.5 + 23.8i)11-s + 36·12-s + (−0.802 + 2.46i)13-s + (103. − 75.4i)14-s + (45.6 + 33.1i)15-s + (−4.94 − 15.2i)16-s + (4.39 + 13.5i)17-s + ⋯ |
| L(s) = 1 | + (0.488 − 1.50i)2-s + (−0.467 + 0.339i)3-s + (−1.21 − 0.881i)4-s + (−0.519 − 1.59i)5-s + (0.282 + 0.868i)6-s + (1.25 + 0.911i)7-s + (−0.639 + 0.464i)8-s + (0.103 − 0.317i)9-s − 2.65·10-s + (0.755 + 0.654i)11-s + 0.866·12-s + (−0.0171 + 0.0526i)13-s + (1.98 − 1.44i)14-s + (0.785 + 0.570i)15-s + (−0.0772 − 0.237i)16-s + (0.0627 + 0.193i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.588741 - 1.21432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.588741 - 1.21432i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-27.5 - 23.8i)T \) |
| good | 2 | \( 1 + (-1.38 + 4.25i)T + (-6.47 - 4.70i)T^{2} \) |
| 5 | \( 1 + (5.80 + 17.8i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (-23.2 - 16.8i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (0.802 - 2.46i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-4.39 - 13.5i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-8.24 + 5.98i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + 86.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-120. - 87.3i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (24.9 - 76.6i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-187. - 136. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (66.3 - 48.2i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 60.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + (89.8 - 65.2i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-101. + 313. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (563. + 409. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (82.0 + 252. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 - 664.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-271. - 834. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (438. + 318. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (330. - 1.01e3i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (222. + 683. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-257. + 791. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−15.77829682916566576344247906505, −14.37170168631153336189668350841, −12.68553410821426070282263092834, −12.03141292043350120342321824481, −11.33346874425822489471502713512, −9.630306165396462956165125023871, −8.424983302359743157101804638660, −5.14045627139769617922465362099, −4.31363659325680666640310738387, −1.47964404028776953856935953483,
4.18133559360769541514077159739, 6.08992759903448719105875703499, 7.20378303211091615818816883884, 7.992817190709516680826597435467, 10.70044460939170448402830645241, 11.62576974985343586017989264488, 13.83092139898099736896727737678, 14.30617431095416082766392949879, 15.28969286528008829289977215105, 16.59320882335285570234240643764
