| L(s) = 1 | + (−1.02 + 1.71i)2-s + (−1.88 − 3.52i)4-s − 6.98i·5-s − 2.64i·7-s + (7.99 + 0.383i)8-s + (11.9 + 7.18i)10-s + 6.55·11-s + 8.12i·13-s + (4.53 + 2.71i)14-s + (−8.86 + 13.3i)16-s + 13.5·17-s − 20.4·19-s + (−24.6 + 13.1i)20-s + (−6.73 + 11.2i)22-s − 20.8i·23-s + ⋯ |
| L(s) = 1 | + (−0.513 + 0.857i)2-s + (−0.472 − 0.881i)4-s − 1.39i·5-s − 0.377i·7-s + (0.998 + 0.0478i)8-s + (1.19 + 0.718i)10-s + 0.596·11-s + 0.625i·13-s + (0.324 + 0.194i)14-s + (−0.554 + 0.832i)16-s + 0.798·17-s − 1.07·19-s + (−1.23 + 0.659i)20-s + (−0.306 + 0.511i)22-s − 0.907i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0478 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0478 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9040916461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9040916461\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.02 - 1.71i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 + 6.98iT - 25T^{2} \) |
| 11 | \( 1 - 6.55T + 121T^{2} \) |
| 13 | \( 1 - 8.12iT - 169T^{2} \) |
| 17 | \( 1 - 13.5T + 289T^{2} \) |
| 19 | \( 1 + 20.4T + 361T^{2} \) |
| 23 | \( 1 + 20.8iT - 529T^{2} \) |
| 29 | \( 1 + 54.4iT - 841T^{2} \) |
| 31 | \( 1 + 38.6iT - 961T^{2} \) |
| 37 | \( 1 - 46.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 25.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 78.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 8.60iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 48.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 63.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 16.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 15.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 31.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 59.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 134.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 155.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 104.T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04170347925952207126375686066, −9.505977263662561027940568773764, −8.471307709141126331919800121208, −8.084000119260267899344604293475, −6.77085318183781793256436203345, −5.97222380117776742184168170078, −4.77268917092446864898204471646, −4.15389685441221418797860724534, −1.69757488231914247677463862228, −0.44442416274627938354814124046,
1.65739169403291088978776917828, 3.00535131820410742695905965861, 3.61934090124653438705084438990, 5.21626670013303441493715997008, 6.59392080140597712879050526908, 7.37285476556094765037856632442, 8.422540473928883878127702709576, 9.296955817334344361277818818591, 10.38228371183243124480478321387, 10.69150592211172361523201196590
