L(s) = 1 | + (−2.52 + 1.27i)2-s + (5.18 + 0.355i)3-s + (4.74 − 6.44i)4-s + (−1.23 + 0.715i)5-s + (−13.5 + 5.71i)6-s + (23.8 + 13.7i)7-s + (−3.76 + 22.3i)8-s + (26.7 + 3.68i)9-s + (2.21 − 3.38i)10-s + (−11.1 + 19.2i)11-s + (26.8 − 31.7i)12-s + (−34.5 − 59.9i)13-s + (−77.8 − 4.33i)14-s + (−6.67 + 3.26i)15-s + (−18.9 − 61.1i)16-s − 31.4i·17-s + ⋯ |
L(s) = 1 | + (−0.892 + 0.451i)2-s + (0.997 + 0.0683i)3-s + (0.593 − 0.805i)4-s + (−0.110 + 0.0639i)5-s + (−0.921 + 0.388i)6-s + (1.28 + 0.744i)7-s + (−0.166 + 0.986i)8-s + (0.990 + 0.136i)9-s + (0.0700 − 0.107i)10-s + (−0.304 + 0.527i)11-s + (0.646 − 0.762i)12-s + (−0.738 − 1.27i)13-s + (−1.48 − 0.0828i)14-s + (−0.114 + 0.0562i)15-s + (−0.296 − 0.955i)16-s − 0.448i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.14732 + 0.370345i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14732 + 0.370345i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.52 - 1.27i)T \) |
| 3 | \( 1 + (-5.18 - 0.355i)T \) |
good | 5 | \( 1 + (1.23 - 0.715i)T + (62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (-23.8 - 13.7i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (11.1 - 19.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (34.5 + 59.9i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 31.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 11.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (72.6 + 125. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (93.6 + 54.0i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (102. - 59.3i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-344. + 199. i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-173. - 100. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (151. - 262. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 243. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (41.9 + 72.6i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-199. + 345. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (307. - 177. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 64.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-354. - 204. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-79.8 + 138. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.49e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-700. + 1.21e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.82760282236134852113391834253, −15.00029734870511690501046489555, −14.31144009482200591153583751513, −12.41515087261441104346291662311, −10.79895714838490830949908382857, −9.523858942469713107600218087155, −8.253052561785771743379805323866, −7.49482998315104966781236146383, −5.19725742416221891397775755261, −2.22479002996772461703893723506,
1.85488749605626474199913172940, 4.04893392898809410912145229624, 7.29185466463485211408010671890, 8.174828041542768363089803534115, 9.413292130602089575019265292180, 10.75963574502542049150516088910, 11.97015284490549748817398626630, 13.56905444583349277904484873923, 14.59061641362701669889543403904, 16.01297496034819337684455059957