L(s) = 1 | + (−2.59 + 1.49i)2-s + (0.478 − 0.828i)4-s + (−7.80 + 13.5i)5-s + (−13.5 + 12.6i)7-s − 21.0i·8-s − 46.6i·10-s + (−46.4 + 26.8i)11-s + (29.1 + 16.8i)13-s + (16.1 − 53.0i)14-s + (35.3 + 61.2i)16-s + 43.5·17-s + 32.2i·19-s + (7.46 + 12.9i)20-s + (80.2 − 139. i)22-s + (−129. − 74.6i)23-s + ⋯ |
L(s) = 1 | + (−0.916 + 0.529i)2-s + (0.0598 − 0.103i)4-s + (−0.697 + 1.20i)5-s + (−0.731 + 0.682i)7-s − 0.931i·8-s − 1.47i·10-s + (−1.27 + 0.735i)11-s + (0.621 + 0.358i)13-s + (0.309 − 1.01i)14-s + (0.552 + 0.957i)16-s + 0.621·17-s + 0.389i·19-s + (0.0834 + 0.144i)20-s + (0.778 − 1.34i)22-s + (−1.17 − 0.676i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0889065 - 0.0726572i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0889065 - 0.0726572i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (13.5 - 12.6i)T \) |
good | 2 | \( 1 + (2.59 - 1.49i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (7.80 - 13.5i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (46.4 - 26.8i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-29.1 - 16.8i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 43.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (129. + 74.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-127. + 73.5i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-61.1 - 35.3i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 40.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-179. + 310. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (253. + 439. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-228. - 395. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 213. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (159. - 276. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (303. - 175. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (289. - 501. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 787. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 146. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (193. + 334. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (98.9 + 171. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 596.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (631. - 364. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.67774307432817828299699412451, −11.93321759394412028356760038552, −10.46374953038948027932258753252, −10.00683854898462861679177553181, −8.680399221117899212811501405809, −7.78614940168832916922266373052, −6.97940945175728378857644344944, −5.96090991911892809516517304955, −3.94824832190599560847897186473, −2.68927454701355912445353775758,
0.084335599405552886186600351381, 1.05774857713269062495389213293, 3.16332088175482854317054390076, 4.71361425728126573815610736953, 5.90444273378786853472437125263, 7.85148499432440239331585414028, 8.265880584201826646201596719181, 9.409836900816564604979021799783, 10.27917390070253450524636144684, 11.12772635728619897037057162998