L(s) = 1 | + (2.17 + 2.06i)3-s + (−3.17 − 3.17i)5-s − 6.03i·7-s + (0.485 + 8.98i)9-s + (−13.0 − 13.0i)11-s + (−6.39 − 6.39i)13-s + (−0.363 − 13.4i)15-s − 4.39i·17-s + (3.21 + 3.21i)19-s + (12.4 − 13.1i)21-s − 34.0·23-s − 4.78i·25-s + (−17.4 + 20.5i)27-s + (27.9 − 27.9i)29-s − 7.90·31-s + ⋯ |
L(s) = 1 | + (0.725 + 0.687i)3-s + (−0.635 − 0.635i)5-s − 0.862i·7-s + (0.0539 + 0.998i)9-s + (−1.18 − 1.18i)11-s + (−0.491 − 0.491i)13-s + (−0.0242 − 0.898i)15-s − 0.258i·17-s + (0.168 + 0.168i)19-s + (0.593 − 0.626i)21-s − 1.47·23-s − 0.191i·25-s + (−0.647 + 0.761i)27-s + (0.964 − 0.964i)29-s − 0.255·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.664037 - 0.880139i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.664037 - 0.880139i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.17 - 2.06i)T \) |
good | 5 | \( 1 + (3.17 + 3.17i)T + 25iT^{2} \) |
| 7 | \( 1 + 6.03iT - 49T^{2} \) |
| 11 | \( 1 + (13.0 + 13.0i)T + 121iT^{2} \) |
| 13 | \( 1 + (6.39 + 6.39i)T + 169iT^{2} \) |
| 17 | \( 1 + 4.39iT - 289T^{2} \) |
| 19 | \( 1 + (-3.21 - 3.21i)T + 361iT^{2} \) |
| 23 | \( 1 + 34.0T + 529T^{2} \) |
| 29 | \( 1 + (-27.9 + 27.9i)T - 841iT^{2} \) |
| 31 | \( 1 + 7.90T + 961T^{2} \) |
| 37 | \( 1 + (20.0 - 20.0i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 45.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36.0 + 36.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 5.08iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-20.7 - 20.7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (39.0 + 39.0i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-49.8 - 49.8i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (44.9 + 44.9i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 46.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 97.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 40.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + (35.5 - 35.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 69.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 61.0T + 9.40e3T^{2} \) |
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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
−10.53854311519583750669275218166, −10.14829392578368226267085308923, −8.882947232849111876166631771878, −8.006751642583810885366319374821, −7.61978345045776287341134409025, −5.79951885081004383572364303181, −4.68800693792824858188491554327, −3.82137417858184122766933607588, −2.65446769704493821629830630552, −0.42102153555175260934399637179,
2.05557486707129818188285189151, 2.90487998424246838809065856788, 4.30199506629665709918941377610, 5.72278868590466788597939720461, 7.01517170665722758581017148934, 7.56758018373411159188483578439, 8.471842844610777663989788410000, 9.486142747058272274362178045458, 10.40108360827325368637125327791, 11.62212158317460070566640854581