L(s) = 1 | + (−0.549 − 1.69i)2-s + (−0.938 + 0.681i)4-s + (0.858 − 2.64i)5-s + (0.809 − 0.587i)7-s + (−1.20 − 0.878i)8-s − 4.93·10-s + (−1.14 + 3.11i)11-s + (−1.32 − 4.08i)13-s + (−1.43 − 1.04i)14-s + (−1.53 + 4.73i)16-s + (0.851 − 2.62i)17-s + (−1.56 − 1.13i)19-s + (0.995 + 3.06i)20-s + (5.89 + 0.232i)22-s − 4.37·23-s + ⋯ |
L(s) = 1 | + (−0.388 − 1.19i)2-s + (−0.469 + 0.340i)4-s + (0.383 − 1.18i)5-s + (0.305 − 0.222i)7-s + (−0.427 − 0.310i)8-s − 1.56·10-s + (−0.346 + 0.938i)11-s + (−0.367 − 1.13i)13-s + (−0.384 − 0.279i)14-s + (−0.384 + 1.18i)16-s + (0.206 − 0.635i)17-s + (−0.359 − 0.261i)19-s + (0.222 + 0.684i)20-s + (1.25 + 0.0495i)22-s − 0.911·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.209335 + 1.03437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.209335 + 1.03437i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (1.14 - 3.11i)T \) |
good | 2 | \( 1 + (0.549 + 1.69i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (-0.858 + 2.64i)T + (-4.04 - 2.93i)T^{2} \) |
| 13 | \( 1 + (1.32 + 4.08i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.851 + 2.62i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.56 + 1.13i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 4.37T + 23T^{2} \) |
| 29 | \( 1 + (-6.98 + 5.07i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.0619 + 0.190i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.837 - 0.608i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.77 - 5.64i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + (10.5 + 7.66i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.20 - 3.70i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (6.92 - 5.02i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.305 - 0.940i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 5.41T + 67T^{2} \) |
| 71 | \( 1 + (-0.623 + 1.91i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.06 + 5.86i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.94 - 5.98i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.531 + 1.63i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + (-3.58 - 11.0i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.989349614340929795246666846118, −9.472173828581678450469462193053, −8.457459015441587314599834390456, −7.69501545349827244292485992588, −6.31663346991576764076284008356, −5.13838205113702105901761391352, −4.38409746298775644446885098699, −2.90224228555753674183278225840, −1.84059808051410924706269074777, −0.61112569032517841190909038833,
2.19281909157309866997415209883, 3.34012362545872837336299739906, 4.89823285178375070822754521596, 6.13784190443090454553493728250, 6.40368922949805037066493297186, 7.40633883620608114704894320586, 8.218725778567875352159981054974, 8.956849225994281091536681853049, 9.996884214673691180903849129730, 10.82987550575137351847332126477