L(s) = 1 | + (−1.87 + 0.684i)2-s + (−0.520 + 2.95i)3-s + (3.06 − 2.57i)4-s + (5.04 + 4.23i)5-s + (−1.04 − 5.90i)6-s + (6.29 + 10.9i)7-s + (−4.00 + 6.92i)8-s + (−8.45 − 3.07i)9-s + (−12.3 − 4.50i)10-s + (23.6 − 41.0i)11-s + (6.00 + 10.3i)12-s + (13.8 + 78.6i)13-s + (−19.3 − 16.1i)14-s + (−15.1 + 12.7i)15-s + (2.77 − 15.7i)16-s + (−75.9 + 27.6i)17-s + ⋯ |
L(s) = 1 | + (−0.664 + 0.241i)2-s + (−0.100 + 0.568i)3-s + (0.383 − 0.321i)4-s + (0.451 + 0.378i)5-s + (−0.0708 − 0.402i)6-s + (0.340 + 0.589i)7-s + (−0.176 + 0.306i)8-s + (−0.313 − 0.114i)9-s + (−0.391 − 0.142i)10-s + (0.649 − 1.12i)11-s + (0.144 + 0.250i)12-s + (0.295 + 1.67i)13-s + (−0.368 − 0.309i)14-s + (−0.260 + 0.218i)15-s + (0.0434 − 0.246i)16-s + (−1.08 + 0.394i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.578764 + 0.940399i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.578764 + 0.940399i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.87 - 0.684i)T \) |
| 3 | \( 1 + (0.520 - 2.95i)T \) |
| 19 | \( 1 + (7.87 - 82.4i)T \) |
good | 5 | \( 1 + (-5.04 - 4.23i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-6.29 - 10.9i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-23.6 + 41.0i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-13.8 - 78.6i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (75.9 - 27.6i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 23 | \( 1 + (164. - 137. i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-68.3 - 24.8i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-106. - 184. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 1.68T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-83.9 + 475. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-268. - 225. i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (129. + 46.9i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 + (74.5 - 62.5i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-780. + 284. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (23.1 - 19.4i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (313. + 114. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-340. - 285. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + (-34.4 + 195. i)T + (-3.65e5 - 1.33e5i)T^{2} \) |
| 79 | \( 1 + (175. - 992. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (194. + 336. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (37.5 + 212. i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-1.31e3 + 477. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−13.98897825978303779458220716869, −11.96155430114079334783287475161, −11.26095027703319220575708023133, −10.17391019809636294273023069383, −9.089241285571335246175524453640, −8.387160937709031029789569334598, −6.59365767497689186731803366612, −5.80432703422785263593451254351, −3.97603298270440526561856397116, −1.94185818976593470987074961630,
0.76144986534737471479951638700, 2.33931479037805307247398081215, 4.50543668545129912255538494108, 6.21477409201739042237699613480, 7.39149189893979670640381405220, 8.395865407518702710790560672598, 9.610351726907170868403053114481, 10.59467003066518620669256853008, 11.68098498682805780565219080393, 12.78814398061536098091423784483