L(s) = 1 | + (−2.43 + 4.21i)2-s + (−7.86 − 13.6i)4-s + 6.21·5-s + (−4.64 + 17.9i)7-s + 37.6·8-s + (−15.1 + 26.2i)10-s − 55.5·11-s + (−6.74 + 11.6i)13-s + (−64.3 − 63.2i)14-s + (−28.7 + 49.8i)16-s + (43.1 − 74.7i)17-s + (−42.1 − 72.9i)19-s + (−48.8 − 84.6i)20-s + (135. − 234. i)22-s − 10.5·23-s + ⋯ |
L(s) = 1 | + (−0.861 + 1.49i)2-s + (−0.983 − 1.70i)4-s + 0.555·5-s + (−0.250 + 0.968i)7-s + 1.66·8-s + (−0.478 + 0.828i)10-s − 1.52·11-s + (−0.143 + 0.249i)13-s + (−1.22 − 1.20i)14-s + (−0.449 + 0.779i)16-s + (0.616 − 1.06i)17-s + (−0.508 − 0.880i)19-s + (−0.546 − 0.945i)20-s + (1.31 − 2.27i)22-s − 0.0955·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.168298 - 0.0794934i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.168298 - 0.0794934i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (4.64 - 17.9i)T \) |
good | 2 | \( 1 + (2.43 - 4.21i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 6.21T + 125T^{2} \) |
| 11 | \( 1 + 55.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + (6.74 - 11.6i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-43.1 + 74.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (42.1 + 72.9i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + 10.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + (76.2 + 132. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-127. - 220. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (172. + 298. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-57.9 + 100. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-46.7 - 80.9i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-159. + 276. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-136. + 236. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-175. - 303. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-145. + 252. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (416. + 721. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 771.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (140. - 242. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (172. - 299. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-62.9 - 108. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-163. - 282. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-124. - 215. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.01678030053966005238376490041, −10.49809750592364922163385802497, −9.609632043042324743561983769335, −8.853832588397808209605580606924, −7.85887242407420343557152845259, −6.88755312592433811058527112955, −5.71434645246407336990415415479, −5.12287885141299540455841109349, −2.49250414381427277291965406225, −0.10800472214479018477384727608,
1.50144938728638796403093130144, 2.85704669811743090696533076401, 4.10172223566009184079194366183, 5.83489182337424146041494978582, 7.63533450855889577907160082883, 8.368595401319113133631919356450, 9.819215742403758584493046927609, 10.25715730471421737830380638106, 10.87168896308018858379660205233, 12.15979861024174693660020780297