L(s) = 1 | + (0.195 + 0.112i)2-s + (−4.76 − 2.06i)3-s + (−3.97 − 6.88i)4-s + (−0.697 − 0.940i)6-s + (27.0 + 15.5i)7-s − 3.59i·8-s + (18.4 + 19.7i)9-s + (−9.06 + 15.6i)11-s + (4.71 + 41.0i)12-s + (43.4 − 25.0i)13-s + (3.51 + 6.08i)14-s + (−31.3 + 54.3i)16-s − 131. i·17-s + (1.37 + 5.92i)18-s − 23.2·19-s + ⋯ |
L(s) = 1 | + (0.0689 + 0.0398i)2-s + (−0.917 − 0.397i)3-s + (−0.496 − 0.860i)4-s + (−0.0474 − 0.0639i)6-s + (1.45 + 0.842i)7-s − 0.158i·8-s + (0.683 + 0.730i)9-s + (−0.248 + 0.430i)11-s + (0.113 + 0.987i)12-s + (0.926 − 0.535i)13-s + (0.0670 + 0.116i)14-s + (−0.490 + 0.849i)16-s − 1.87i·17-s + (0.0180 + 0.0775i)18-s − 0.280·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.967006 - 0.862463i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.967006 - 0.862463i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.76 + 2.06i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.195 - 0.112i)T + (4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (-27.0 - 15.5i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (9.06 - 15.6i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-43.4 + 25.0i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 131. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 23.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-28.5 + 16.4i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-62.9 + 108. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (62.5 + 108. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 99.9iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (122. + 212. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-120. - 69.5i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-409. - 236. i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 421. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (371. + 642. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (4.48 - 7.77i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-510. + 294. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 48.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 409. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-265. + 459. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (255. + 147. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 852.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-336. - 194. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.41614385191418488019273156404, −10.89169074126209429372592611454, −9.729899516012709357664162295320, −8.590606253146384596437514290568, −7.50903180177448337072278312534, −6.14237739723586669061279002900, −5.24077799284999384940504580433, −4.64402377376472723693588867738, −2.06019081545897980338826365704, −0.70711859478107532151503729046,
1.29848408443658790696023179262, 3.79215585467526443981267857930, 4.43022947892390594481761696625, 5.62491485424544397421402050278, 7.00159039982245476742946252985, 8.153673318723225459616870781812, 8.885232023074609074832461077326, 10.55651043340790827185241176927, 10.94591126778425063934239985473, 11.91954794185540660858506225741