L(s) = 1 | + (−3.15 − 1.02i)2-s + (−2.42 + 1.76i)3-s + (5.66 + 4.11i)4-s + (−6.30 + 2.04i)5-s + (9.46 − 3.07i)6-s + (6.47 + 4.70i)7-s + (−5.84 − 8.04i)8-s + (2.78 − 8.55i)9-s + 22·10-s − 21.0·12-s + (1.23 − 3.80i)13-s + (−15.5 − 21.4i)14-s + (11.6 − 16.0i)15-s + (1.54 + 4.75i)16-s + (12.6 − 4.09i)17-s + (−17.5 + 24.1i)18-s + ⋯ |
L(s) = 1 | + (−1.57 − 0.512i)2-s + (−0.809 + 0.587i)3-s + (1.41 + 1.02i)4-s + (−1.26 + 0.409i)5-s + (1.57 − 0.512i)6-s + (0.924 + 0.671i)7-s + (−0.731 − 1.00i)8-s + (0.309 − 0.951i)9-s + 2.20·10-s − 1.75·12-s + (0.0950 − 0.292i)13-s + (−1.11 − 1.53i)14-s + (0.779 − 1.07i)15-s + (0.0965 + 0.297i)16-s + (0.742 − 0.241i)17-s + (−0.974 + 1.34i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.343978 + 0.237630i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.343978 + 0.237630i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.42 - 1.76i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (3.15 + 1.02i)T + (3.23 + 2.35i)T^{2} \) |
| 5 | \( 1 + (6.30 - 2.04i)T + (20.2 - 14.6i)T^{2} \) |
| 7 | \( 1 + (-6.47 - 4.70i)T + (15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (-1.23 + 3.80i)T + (-136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (-12.6 + 4.09i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-4.85 + 3.52i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 + 6.63iT - 529T^{2} \) |
| 29 | \( 1 + (-23.3 + 32.1i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (8.03 - 24.7i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (24.2 + 17.6i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-7.79 - 10.7i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 - 42T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-50.6 - 69.7i)T + (-682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-56.7 - 18.4i)T + (2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (38.9 - 53.6i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-3.70 - 11.4i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 - 2T + 4.48e3T^{2} \) |
| 71 | \( 1 + (56.7 - 18.4i)T + (4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-59.8 - 43.4i)T + (1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (12.3 - 38.0i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (37.8 - 12.2i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 - 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-19.1 + 58.9i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−11.07209019237892392186902098127, −10.68328458387575020849053872457, −9.634012372013799375906264606849, −8.679447536484033300550180387762, −7.88341770562719645103318342451, −7.07647594201941075648239784983, −5.56376522737039599327894680255, −4.25556373272528872102164947909, −2.83544028863545216416533404595, −0.938393870598518989578418316874,
0.52240310088701765853950783297, 1.52787987289043795210358224966, 4.12601074968001562683458886930, 5.38473843301690949403023677808, 6.75805430514889041481350434862, 7.57735612464915749519789026980, 7.963569580685891511343164232848, 8.868237129343867902983123779522, 10.25320408211992432785583603706, 10.90154957648555544256049865657