L(s) = 1 | + (−4.49 − 1.45i)2-s + (−4.20 + 3.05i)3-s + (5.11 + 3.71i)4-s + (5.35 + 16.4i)5-s + (23.3 − 7.58i)6-s + (48.2 − 66.3i)7-s + (26.8 + 37.0i)8-s + (8.34 − 25.6i)9-s − 81.9i·10-s + (120. − 2.78i)11-s − 32.8·12-s + (162. + 52.6i)13-s + (−313. + 227. i)14-s + (−72.8 − 52.9i)15-s + (−98.0 − 301. i)16-s + (−314. + 102. i)17-s + ⋯ |
L(s) = 1 | + (−1.12 − 0.364i)2-s + (−0.467 + 0.339i)3-s + (0.319 + 0.232i)4-s + (0.214 + 0.659i)5-s + (0.648 − 0.210i)6-s + (0.984 − 1.35i)7-s + (0.420 + 0.578i)8-s + (0.103 − 0.317i)9-s − 0.819i·10-s + (0.999 − 0.0230i)11-s − 0.227·12-s + (0.959 + 0.311i)13-s + (−1.60 + 1.16i)14-s + (−0.323 − 0.235i)15-s + (−0.382 − 1.17i)16-s + (−1.08 + 0.353i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.784385 - 0.123804i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.784385 - 0.123804i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.20 - 3.05i)T \) |
| 11 | \( 1 + (-120. + 2.78i)T \) |
good | 2 | \( 1 + (4.49 + 1.45i)T + (12.9 + 9.40i)T^{2} \) |
| 5 | \( 1 + (-5.35 - 16.4i)T + (-505. + 367. i)T^{2} \) |
| 7 | \( 1 + (-48.2 + 66.3i)T + (-741. - 2.28e3i)T^{2} \) |
| 13 | \( 1 + (-162. - 52.6i)T + (2.31e4 + 1.67e4i)T^{2} \) |
| 17 | \( 1 + (314. - 102. i)T + (6.75e4 - 4.90e4i)T^{2} \) |
| 19 | \( 1 + (-159. - 219. i)T + (-4.02e4 + 1.23e5i)T^{2} \) |
| 23 | \( 1 - 414.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-631. + 868. i)T + (-2.18e5 - 6.72e5i)T^{2} \) |
| 31 | \( 1 + (-167. + 516. i)T + (-7.47e5 - 5.42e5i)T^{2} \) |
| 37 | \( 1 + (508. + 369. i)T + (5.79e5 + 1.78e6i)T^{2} \) |
| 41 | \( 1 + (144. + 198. i)T + (-8.73e5 + 2.68e6i)T^{2} \) |
| 43 | \( 1 - 3.52e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (1.90e3 - 1.38e3i)T + (1.50e6 - 4.64e6i)T^{2} \) |
| 53 | \( 1 + (-316. + 973. i)T + (-6.38e6 - 4.63e6i)T^{2} \) |
| 59 | \( 1 + (1.75e3 + 1.27e3i)T + (3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 + (-3.67e3 + 1.19e3i)T + (1.12e7 - 8.13e6i)T^{2} \) |
| 67 | \( 1 + 70.0T + 2.01e7T^{2} \) |
| 71 | \( 1 + (141. + 435. i)T + (-2.05e7 + 1.49e7i)T^{2} \) |
| 73 | \( 1 + (3.91e3 - 5.38e3i)T + (-8.77e6 - 2.70e7i)T^{2} \) |
| 79 | \( 1 + (-7.78e3 - 2.52e3i)T + (3.15e7 + 2.28e7i)T^{2} \) |
| 83 | \( 1 + (-301. + 98.1i)T + (3.83e7 - 2.78e7i)T^{2} \) |
| 89 | \( 1 + 5.30e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (1.62e3 - 5.00e3i)T + (-7.16e7 - 5.20e7i)T^{2} \) |
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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
−16.42194333986839336440328690523, −14.61098188511678534006277867902, −13.71245155934525230703499835352, −11.33007371587206851693159099326, −10.89058780498468468243664746822, −9.728094151679720436886732553641, −8.251781887550054190870764096536, −6.67071322382918047098921856573, −4.34178479717416912650465240982, −1.22666303533443917939449221089,
1.31499547195132819186958753782, 5.06955638051269447172872679153, 6.79149516157012544527593153888, 8.617506048004414928093891714084, 8.985924625887846365185142385125, 10.96948149142896441870002279276, 12.15247836883124457066365738173, 13.48387308269842707189971480089, 15.25700513410845911216684943023, 16.31369804719278110062457846970