L(s) = 1 | + (−4.63 − 4.63i)3-s + (29.2 + 29.2i)5-s − 59.6·7-s − 38.0i·9-s + (−18.0 + 18.0i)11-s + (−50.7 + 50.7i)13-s − 270. i·15-s − 223.·17-s + (14.7 + 14.7i)19-s + (276. + 276. i)21-s − 739.·23-s + 1.08e3i·25-s + (−551. + 551. i)27-s + (−938. + 938. i)29-s + 938. i·31-s + ⋯ |
L(s) = 1 | + (−0.515 − 0.515i)3-s + (1.16 + 1.16i)5-s − 1.21·7-s − 0.469i·9-s + (−0.149 + 0.149i)11-s + (−0.300 + 0.300i)13-s − 1.20i·15-s − 0.774·17-s + (0.0408 + 0.0408i)19-s + (0.626 + 0.626i)21-s − 1.39·23-s + 1.72i·25-s + (−0.756 + 0.756i)27-s + (−1.11 + 1.11i)29-s + 0.976i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.117794 + 0.415962i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.117794 + 0.415962i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (4.63 + 4.63i)T + 81iT^{2} \) |
| 5 | \( 1 + (-29.2 - 29.2i)T + 625iT^{2} \) |
| 7 | \( 1 + 59.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + (18.0 - 18.0i)T - 1.46e4iT^{2} \) |
| 13 | \( 1 + (50.7 - 50.7i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + 223.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (-14.7 - 14.7i)T + 1.30e5iT^{2} \) |
| 23 | \( 1 + 739.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (938. - 938. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 - 938. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (263. + 263. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 - 248. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.03e3 + 1.03e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.01e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (833. + 833. i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 + (2.22e3 - 2.22e3i)T - 1.21e7iT^{2} \) |
| 61 | \( 1 + (-341. + 341. i)T - 1.38e7iT^{2} \) |
| 67 | \( 1 + (-4.84e3 - 4.84e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 - 4.18e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 9.07e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 735. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-1.44e3 - 1.44e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + 5.07e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.52e3T + 8.85e7T^{2} \) |
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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
−13.04337276552230574813275571752, −12.17095821325186965330150302098, −10.89366032364436719629283882571, −9.977982073491203186661309803249, −9.175271463370932557492348637940, −7.09943836234043043507240858683, −6.53429024408288252148542384815, −5.67748178742977164595775841292, −3.44164940185344518673698166186, −2.04063070418753170461591560135,
0.17491578382766491059102145480, 2.21701928078806766835142824230, 4.27629832461966334730737585261, 5.50894796676766502277863581628, 6.22050893374087887026501686549, 8.050152981934790991719778205600, 9.528669764683825544916629033439, 9.783078150896455217902540418910, 11.05274116441837768228067013393, 12.44499242619513407818966742899