L(s) = 1 | + (−1.76 + 1.02i)2-s + (−1.5 − 2.59i)3-s + (−1.91 + 3.31i)4-s − 12.0i·5-s + (5.30 + 3.06i)6-s + (−25.7 − 14.8i)7-s − 24.1i·8-s + (−4.5 + 7.79i)9-s + (12.3 + 21.3i)10-s + (24.3 − 14.0i)11-s + 11.4·12-s + (−40.9 + 22.7i)13-s + 60.7·14-s + (−31.3 + 18.1i)15-s + (9.35 + 16.1i)16-s + (−25.3 + 43.8i)17-s + ⋯ |
L(s) = 1 | + (−0.625 + 0.361i)2-s + (−0.288 − 0.499i)3-s + (−0.239 + 0.414i)4-s − 1.08i·5-s + (0.361 + 0.208i)6-s + (−1.39 − 0.802i)7-s − 1.06i·8-s + (−0.166 + 0.288i)9-s + (0.390 + 0.675i)10-s + (0.666 − 0.384i)11-s + 0.276·12-s + (−0.874 + 0.485i)13-s + 1.15·14-s + (−0.540 + 0.311i)15-s + (0.146 + 0.253i)16-s + (−0.361 + 0.625i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.851i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.205064 - 0.367485i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.205064 - 0.367485i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 + 2.59i)T \) |
| 13 | \( 1 + (40.9 - 22.7i)T \) |
good | 2 | \( 1 + (1.76 - 1.02i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + 12.0iT - 125T^{2} \) |
| 7 | \( 1 + (25.7 + 14.8i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-24.3 + 14.0i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (25.3 - 43.8i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-91.0 - 52.5i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (80.2 + 139. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (70.0 + 121. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 223. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (197. - 114. i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-256. + 147. i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-96.0 + 166. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 36.9iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (380. + 219. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (143. - 247. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-465. + 268. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-88.9 - 51.3i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 75.5iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 17.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.46e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (290. - 167. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (648. + 374. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−16.01252664609319720183808427469, −13.86002389789380273452267299145, −12.84746585774451996722514172268, −12.11138515289537244360122868811, −9.975388262854962172674348896250, −8.988165782113877105596535018450, −7.60353479332672980997296528959, −6.34583066508708487404185016338, −4.04906305147848961248537148533, −0.44252120157333791241515461499,
2.96287965405700972635709823728, 5.46765559899408091864602775538, 6.96451625399292103968388608311, 9.297815221422881554592351839050, 9.784188708236655230519418990179, 11.01916085426739010892371408319, 12.22757587531071911565928937877, 14.01291221112426582029265757375, 15.09169724361594859957798558197, 16.03317019471565696270976830159