L(s) = 1 | + (6.67 − 11.5i)5-s + (7.17 + 12.4i)7-s + (19.5 + 33.8i)11-s + (38.3 − 66.5i)13-s + 62.4·17-s − 39.7·19-s + (−64.2 + 111. i)23-s + (−26.7 − 46.2i)25-s + (32.4 + 56.2i)29-s + (4.56 − 7.91i)31-s + 191.·35-s + 319.·37-s + (8.78 − 15.2i)41-s + (225. + 390. i)43-s + (−290. − 503. i)47-s + ⋯ |
L(s) = 1 | + (0.597 − 1.03i)5-s + (0.387 + 0.671i)7-s + (0.535 + 0.927i)11-s + (0.819 − 1.41i)13-s + 0.890·17-s − 0.480·19-s + (−0.582 + 1.00i)23-s + (−0.213 − 0.370i)25-s + (0.207 + 0.360i)29-s + (0.0264 − 0.0458i)31-s + 0.926·35-s + 1.41·37-s + (0.0334 − 0.0579i)41-s + (0.799 + 1.38i)43-s + (−0.902 − 1.56i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.631520868\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.631520868\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-6.67 + 11.5i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (-7.17 - 12.4i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-19.5 - 33.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-38.3 + 66.5i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 62.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 39.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + (64.2 - 111. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-32.4 - 56.2i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-4.56 + 7.91i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 319.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-8.78 + 15.2i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-225. - 390. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (290. + 503. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 329.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (120. - 209. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (248. + 430. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-289. + 500. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 660.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 696.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (365. + 632. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-545. - 944. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 317.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-742. - 1.28e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−9.863731900644336436020781672881, −9.340742034230592686913326295808, −8.342164770669378088865033090419, −7.75392953548938847633050847821, −6.26808375427947758207567277258, −5.51184055342439213003453943573, −4.77041989462041670949314571297, −3.45614152052246735214718721968, −1.96839526764580073171407756559, −0.993414654760538544600297941068,
1.05050541809741534273719374652, 2.33529627726776542476107029889, 3.59704686990743989085639066156, 4.47075793541465010321415249741, 6.14698471684070785360939733270, 6.35876032297143470079720294631, 7.49847432326514652915544826537, 8.478656445614642500464345962552, 9.412001803606853433080252711961, 10.33544563745904210346989645491