| L(s) = 1 | + (4.96 + 8.59i)2-s + (−3.07 − 15.2i)3-s + (−33.2 + 57.5i)4-s + (23.6 − 40.9i)5-s + (116. − 102. i)6-s + (1.01 + 1.75i)7-s − 342.·8-s + (−224. + 94.0i)9-s + 469.·10-s + (−91.8 − 159. i)11-s + (981. + 330. i)12-s + (−364. + 631. i)13-s + (−10.0 + 17.4i)14-s + (−698. − 235. i)15-s + (−633. − 1.09e3i)16-s + 1.21e3·17-s + ⋯ |
| L(s) = 1 | + (0.877 + 1.51i)2-s + (−0.197 − 0.980i)3-s + (−1.03 + 1.79i)4-s + (0.423 − 0.732i)5-s + (1.31 − 1.15i)6-s + (0.00783 + 0.0135i)7-s − 1.88·8-s + (−0.922 + 0.387i)9-s + 1.48·10-s + (−0.228 − 0.396i)11-s + (1.96 + 0.663i)12-s + (−0.598 + 1.03i)13-s + (−0.0137 + 0.0237i)14-s + (−0.802 − 0.270i)15-s + (−0.618 − 1.07i)16-s + 1.01·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.30942 + 0.798853i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30942 + 0.798853i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.07 + 15.2i)T \) |
| good | 2 | \( 1 + (-4.96 - 8.59i)T + (-16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-23.6 + 40.9i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-1.01 - 1.75i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (91.8 + 159. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (364. - 631. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 - 1.21e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 473.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.80e3 + 3.12e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-663. - 1.14e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-2.59e3 + 4.48e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 1.47e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (3.15e3 - 5.47e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (3.06e3 + 5.31e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.58e3 + 2.74e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 1.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.47e4 + 2.55e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.01e4 - 3.49e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.17e4 - 2.03e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 123.T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-2.40e4 - 4.16e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (5.16e3 + 8.95e3i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + 4.25e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (4.93e4 + 8.54e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−21.08422922092424332534568171829, −18.87017379577638213109426588617, −17.16621064022825251733483316395, −16.52378437884400633712464100450, −14.54826715978149673188711154130, −13.42611888500311608699194913421, −12.28497831787360153838628019080, −8.498770585085928368271126670211, −6.83932963805136722150960877692, −5.25231378558432049875416255919,
3.12681825814123231437190397886, 5.24881676710935078689420016653, 9.885973724365649413497495278436, 10.71457140431737216659673806269, 12.27545106809466495277730798083, 14.04918090474888995332096533400, 15.20759840568603704754145772406, 17.58584924982977518010294460965, 19.28759002250784227614983518644, 20.60438797494538884104979622324
